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Properties

Label 4-1575e2-1.1-c3e2-0-8
Degree 44
Conductor 24806252480625
Sign 11
Analytic cond. 8635.618635.61
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 4-s − 14·7-s + 3·8-s − 62·11-s + 6·13-s − 42·14-s + 9·16-s + 40·17-s − 122·19-s − 186·22-s + 16·23-s + 18·26-s − 14·28-s − 352·29-s + 66·31-s − 165·32-s + 120·34-s + 188·37-s − 366·38-s − 16·41-s + 396·43-s − 62·44-s + 48·46-s − 188·47-s + 147·49-s + 6·52-s + ⋯
L(s)  = 1  + 1.06·2-s + 1/8·4-s − 0.755·7-s + 0.132·8-s − 1.69·11-s + 0.128·13-s − 0.801·14-s + 9/64·16-s + 0.570·17-s − 1.47·19-s − 1.80·22-s + 0.145·23-s + 0.135·26-s − 0.0944·28-s − 2.25·29-s + 0.382·31-s − 0.911·32-s + 0.605·34-s + 0.835·37-s − 1.56·38-s − 0.0609·41-s + 1.40·43-s − 0.212·44-s + 0.153·46-s − 0.583·47-s + 3/7·49-s + 0.0160·52-s + ⋯

Functional equation

Λ(s)=(2480625s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2480625s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 24806252480625    =    34⋅54⋅723^{4} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 8635.618635.61
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 2480625, ( :3/2,3/2), 1)(4,\ 2480625,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
5 1 1
7C1C_1 (1+pT)2 ( 1 + p T )^{2}
good2D4D_{4} 1−3T+p3T2−3p3T3+p6T4 1 - 3 T + p^{3} T^{2} - 3 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+62T+3582T2+62p3T3+p6T4 1 + 62 T + 3582 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−6T+3378T2−6p3T3+p6T4 1 - 6 T + 3378 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1−40T+8750T2−40p3T3+p6T4 1 - 40 T + 8750 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1+122T+17398T2+122p3T3+p6T4 1 + 122 T + 17398 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−16T+34pT2−16p3T3+p6T4 1 - 16 T + 34 p T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+352T+78278T2+352p3T3+p6T4 1 + 352 T + 78278 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−66T+45870T2−66p3T3+p6T4 1 - 66 T + 45870 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−188T+44542T2−188p3T3+p6T4 1 - 188 T + 44542 T^{2} - 188 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+16T+18350T2+16p3T3+p6T4 1 + 16 T + 18350 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−396T+2226pT2−396p3T3+p6T4 1 - 396 T + 2226 p T^{2} - 396 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+4pT+15582T2+4p4T3+p6T4 1 + 4 p T + 15582 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4}
53D4D_{4} 1−982T+504354T2−982p3T3+p6T4 1 - 982 T + 504354 T^{2} - 982 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+516T+418118T2+516p3T3+p6T4 1 + 516 T + 418118 T^{2} + 516 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1+880T+575238T2+880p3T3+p6T4 1 + 880 T + 575238 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−356T+100374T2−356p3T3+p6T4 1 - 356 T + 100374 T^{2} - 356 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+310T+664038T2+310p3T3+p6T4 1 + 310 T + 664038 T^{2} + 310 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+326T+789802T2+326p3T3+p6T4 1 + 326 T + 789802 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−1832T+1824478T2−1832p3T3+p6T4 1 - 1832 T + 1824478 T^{2} - 1832 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+680T+744870T2+680p3T3+p6T4 1 + 680 T + 744870 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1+796T+411158T2+796p3T3+p6T4 1 + 796 T + 411158 T^{2} + 796 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−670T+1145410T2−670p3T3+p6T4 1 - 670 T + 1145410 T^{2} - 670 p^{3} T^{3} + p^{6} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−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

−8.761543260320092012523303247857, −8.575373559479012227022536871869, −7.84234479147763474799855470839, −7.57878062627741873562892991580, −7.41779908318893452241332078150, −6.75132231929841740465151123716, −6.25544222955419009800857274325, −5.88656040652363805603672663416, −5.48125778991697088863055260271, −5.21806612470637396149750558220, −4.62089319860350269156947609330, −4.26124502480101793559116704086, −3.80202910354351671165325917116, −3.44510499800316644381209374912, −2.73969600391756561547826051309, −2.42003280597919928770959819897, −1.86520601809035991412245784928, −1.00965334270187999610832995518, 0, 0, 1.00965334270187999610832995518, 1.86520601809035991412245784928, 2.42003280597919928770959819897, 2.73969600391756561547826051309, 3.44510499800316644381209374912, 3.80202910354351671165325917116, 4.26124502480101793559116704086, 4.62089319860350269156947609330, 5.21806612470637396149750558220, 5.48125778991697088863055260271, 5.88656040652363805603672663416, 6.25544222955419009800857274325, 6.75132231929841740465151123716, 7.41779908318893452241332078150, 7.57878062627741873562892991580, 7.84234479147763474799855470839, 8.575373559479012227022536871869, 8.761543260320092012523303247857

Graph of the ZZ-function along the critical line