Properties

Label 4-45e2-1.1-c11e2-0-3
Degree 44
Conductor 20252025
Sign 11
Analytic cond. 1195.461195.46
Root an. cond. 5.880085.88008
Motivic weight 1111
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 20·2-s + 1.64e3·4-s + 6.25e3·5-s + 5.79e4·7-s + 9.85e4·8-s + 1.25e5·10-s + 6.18e5·11-s + 3.41e6·13-s + 1.15e6·14-s + 6.29e5·16-s − 1.31e6·17-s + 5.32e6·19-s + 1.02e7·20-s + 1.23e7·22-s − 5.89e7·23-s + 2.92e7·25-s + 6.82e7·26-s + 9.49e7·28-s − 9.41e7·29-s + 2.44e8·31-s + 1.18e8·32-s − 2.63e7·34-s + 3.61e8·35-s + 2.10e7·37-s + 1.06e8·38-s + 6.16e8·40-s + 7.45e8·41-s + ⋯
L(s)  = 1  + 0.441·2-s + 0.800·4-s + 0.894·5-s + 1.30·7-s + 1.06·8-s + 0.395·10-s + 1.15·11-s + 2.55·13-s + 0.575·14-s + 0.150·16-s − 0.225·17-s + 0.493·19-s + 0.716·20-s + 0.511·22-s − 1.90·23-s + 3/5·25-s + 1.12·26-s + 1.04·28-s − 0.852·29-s + 1.53·31-s + 0.622·32-s − 0.0994·34-s + 1.16·35-s + 0.0497·37-s + 0.218·38-s + 0.951·40-s + 1.00·41-s + ⋯

Functional equation

Λ(s)=(2025s/2ΓC(s)2L(s)=(Λ(12s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}
Λ(s)=(2025s/2ΓC(s+11/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20252025    =    34523^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 1195.461195.46
Root analytic conductor: 5.880085.88008
Motivic weight: 1111
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2025, ( :11/2,11/2), 1)(4,\ 2025,\ (\ :11/2, 11/2),\ 1)

Particular Values

L(6)L(6) \approx 10.3986247110.39862471
L(12)L(\frac12) \approx 10.3986247110.39862471
L(132)L(\frac{13}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
5C1C_1 (1p5T)2 ( 1 - p^{5} T )^{2}
good2D4D_{4} 15p2T155p3T25p13T3+p22T4 1 - 5 p^{2} T - 155 p^{3} T^{2} - 5 p^{13} T^{3} + p^{22} T^{4}
7D4D_{4} 157900T+660624350pT257900p11T3+p22T4 1 - 57900 T + 660624350 p T^{2} - 57900 p^{11} T^{3} + p^{22} T^{4}
11D4D_{4} 1618176T+245194892966T2618176p11T3+p22T4 1 - 618176 T + 245194892966 T^{2} - 618176 p^{11} T^{3} + p^{22} T^{4}
13D4D_{4} 13414260T+6398662197390T23414260p11T3+p22T4 1 - 3414260 T + 6398662197390 T^{2} - 3414260 p^{11} T^{3} + p^{22} T^{4}
17D4D_{4} 1+1317940T+59308395866630T2+1317940p11T3+p22T4 1 + 1317940 T + 59308395866630 T^{2} + 1317940 p^{11} T^{3} + p^{22} T^{4}
19D4D_{4} 1280280pT+194538827137638T2280280p12T3+p22T4 1 - 280280 p T + 194538827137638 T^{2} - 280280 p^{12} T^{3} + p^{22} T^{4}
23D4D_{4} 1+2562780pT+2773540471931410T2+2562780p12T3+p22T4 1 + 2562780 p T + 2773540471931410 T^{2} + 2562780 p^{12} T^{3} + p^{22} T^{4}
29D4D_{4} 1+3246220pT+23426350431097358T2+3246220p12T3+p22T4 1 + 3246220 p T + 23426350431097358 T^{2} + 3246220 p^{12} T^{3} + p^{22} T^{4}
31D4D_{4} 1244543464T+34393316207729486T2244543464p11T3+p22T4 1 - 244543464 T + 34393316207729486 T^{2} - 244543464 p^{11} T^{3} + p^{22} T^{4}
37D4D_{4} 121003220T+137126715218410590T221003220p11T3+p22T4 1 - 21003220 T + 137126715218410590 T^{2} - 21003220 p^{11} T^{3} + p^{22} T^{4}
41D4D_{4} 1745743316T+929792912462405846T2745743316p11T3+p22T4 1 - 745743316 T + 929792912462405846 T^{2} - 745743316 p^{11} T^{3} + p^{22} T^{4}
43D4D_{4} 1629950100T+1840945003918927050T2629950100p11T3+p22T4 1 - 629950100 T + 1840945003918927050 T^{2} - 629950100 p^{11} T^{3} + p^{22} T^{4}
47D4D_{4} 11402061540T+5181805952108806370T21402061540p11T3+p22T4 1 - 1402061540 T + 5181805952108806370 T^{2} - 1402061540 p^{11} T^{3} + p^{22} T^{4}
53D4D_{4} 1+1138320580T2203723231625575330T2+1138320580p11T3+p22T4 1 + 1138320580 T - 2203723231625575330 T^{2} + 1138320580 p^{11} T^{3} + p^{22} T^{4}
59D4D_{4} 1+7317515560T+55027608950440780118T2+7317515560p11T3+p22T4 1 + 7317515560 T + 55027608950440780118 T^{2} + 7317515560 p^{11} T^{3} + p^{22} T^{4}
61D4D_{4} 1+1516425676T+85869525433683691566T2+1516425676p11T3+p22T4 1 + 1516425676 T + 85869525433683691566 T^{2} + 1516425676 p^{11} T^{3} + p^{22} T^{4}
67D4D_{4} 115734290140T+ 1 - 15734290140 T + 30 ⁣ ⁣3030\!\cdots\!30T215734290140p11T3+p22T4 T^{2} - 15734290140 p^{11} T^{3} + p^{22} T^{4}
71D4D_{4} 1+32938471544T+ 1 + 32938471544 T + 73 ⁣ ⁣2673\!\cdots\!26T2+32938471544p11T3+p22T4 T^{2} + 32938471544 p^{11} T^{3} + p^{22} T^{4}
73D4D_{4} 1+29982848860T+ 1 + 29982848860 T + 78 ⁣ ⁣1078\!\cdots\!10T2+29982848860p11T3+p22T4 T^{2} + 29982848860 p^{11} T^{3} + p^{22} T^{4}
79D4D_{4} 1+3302823120T+ 1 + 3302823120 T + 12 ⁣ ⁣5812\!\cdots\!58T2+3302823120p11T3+p22T4 T^{2} + 3302823120 p^{11} T^{3} + p^{22} T^{4}
83D4D_{4} 1+13299102420T+ 1 + 13299102420 T + 18 ⁣ ⁣3018\!\cdots\!30T2+13299102420p11T3+p22T4 T^{2} + 13299102420 p^{11} T^{3} + p^{22} T^{4}
89D4D_{4} 112674770860T+ 1 - 12674770860 T + 43 ⁣ ⁣7843\!\cdots\!78T212674770860p11T3+p22T4 T^{2} - 12674770860 p^{11} T^{3} + p^{22} T^{4}
97D4D_{4} 1+3080703740T+ 1 + 3080703740 T + 18 ⁣ ⁣7018\!\cdots\!70T2+3080703740p11T3+p22T4 T^{2} + 3080703740 p^{11} T^{3} + p^{22} T^{4}
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   L(s)=p j=14(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.59106970106290456839370614740, −13.56710128098156843287499157000, −12.55223479424372090912921781289, −11.69758221571725753929748401754, −11.40418902321999187145402067319, −10.90304658380817722644706725155, −10.36095623571164980771795751486, −9.483558352005119969532702819207, −8.773033566273907711214682152651, −8.150800460180734116353941224084, −7.52842368174729526223788513249, −6.50372507392322184884428906988, −6.12918124445120266732680931286, −5.58109298499102332828255387665, −4.32419327711743112445260313694, −4.16813968501628161083361706858, −3.00267797686282288936505338766, −1.83037914699779268756640265577, −1.60277424795465373480873090605, −0.986321171235488763153926762507, 0.986321171235488763153926762507, 1.60277424795465373480873090605, 1.83037914699779268756640265577, 3.00267797686282288936505338766, 4.16813968501628161083361706858, 4.32419327711743112445260313694, 5.58109298499102332828255387665, 6.12918124445120266732680931286, 6.50372507392322184884428906988, 7.52842368174729526223788513249, 8.150800460180734116353941224084, 8.773033566273907711214682152651, 9.483558352005119969532702819207, 10.36095623571164980771795751486, 10.90304658380817722644706725155, 11.40418902321999187145402067319, 11.69758221571725753929748401754, 12.55223479424372090912921781289, 13.56710128098156843287499157000, 13.59106970106290456839370614740

Graph of the ZZ-function along the critical line