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Properties

Label 4-768e2-1.1-c3e2-0-6
Degree 44
Conductor 589824589824
Sign 11
Analytic cond. 2053.312053.31
Root an. cond. 6.731526.73152
Motivic weight 33
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  − 6·3-s + 8·5-s + 16·7-s + 27·9-s − 8·11-s − 72·13-s − 48·15-s − 36·17-s + 136·19-s − 96·21-s + 256·23-s + 6·25-s − 108·27-s − 152·29-s − 80·31-s + 48·33-s + 128·35-s − 136·37-s + 432·39-s − 436·41-s − 712·43-s + 216·45-s − 224·47-s − 286·49-s + 216·51-s − 344·53-s − 64·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.715·5-s + 0.863·7-s + 9-s − 0.219·11-s − 1.53·13-s − 0.826·15-s − 0.513·17-s + 1.64·19-s − 0.997·21-s + 2.32·23-s + 0.0479·25-s − 0.769·27-s − 0.973·29-s − 0.463·31-s + 0.253·33-s + 0.618·35-s − 0.604·37-s + 1.77·39-s − 1.66·41-s − 2.52·43-s + 0.715·45-s − 0.695·47-s − 0.833·49-s + 0.593·51-s − 0.891·53-s − 0.156·55-s + ⋯

Functional equation

Λ(s)=(589824s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(589824s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 589824589824    =    216⋅322^{16} \cdot 3^{2}
Sign: 11
Analytic conductor: 2053.312053.31
Root analytic conductor: 6.731526.73152
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 589824, ( :3/2,3/2), 1)(4,\ 589824,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.0478180631.047818063
L(12)L(\frac12) ≈\approx 1.0478180631.047818063
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)
bad2 1 1
3C1C_1 (1+pT)2 ( 1 + p T )^{2}
good5D4D_{4} 1−8T+58T2−8p3T3+p6T4 1 - 8 T + 58 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4}
7D4D_{4} 1−16T+542T2−16p3T3+p6T4 1 - 16 T + 542 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+8T−650T2+8p3T3+p6T4 1 + 8 T - 650 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+72T+4858T2+72p3T3+p6T4 1 + 72 T + 4858 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+36T+6822T2+36p3T3+p6T4 1 + 36 T + 6822 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−136T+15014T2−136p3T3+p6T4 1 - 136 T + 15014 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1−256T+39886T2−256p3T3+p6T4 1 - 256 T + 39886 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+152T+37706T2+152p3T3+p6T4 1 + 152 T + 37706 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+80T+26030T2+80p3T3+p6T4 1 + 80 T + 26030 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+136T+102602T2+136p3T3+p6T4 1 + 136 T + 102602 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+436T+102166T2+436p3T3+p6T4 1 + 436 T + 102166 T^{2} + 436 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1+712T+282422T2+712p3T3+p6T4 1 + 712 T + 282422 T^{2} + 712 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+224T+179422T2+224p3T3+p6T4 1 + 224 T + 179422 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+344T+280538T2+344p3T3+p6T4 1 + 344 T + 280538 T^{2} + 344 p^{3} T^{3} + p^{6} T^{4}
59C2C_2 (1+324T+p3T2)2 ( 1 + 324 T + p^{3} T^{2} )^{2}
61C2C_2 (1+324T+p3T2)2 ( 1 + 324 T + p^{3} T^{2} )^{2}
67D4D_{4} 1−456T+174278T2−456p3T3+p6T4 1 - 456 T + 174278 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1−2048T+1763566T2−2048p3T3+p6T4 1 - 2048 T + 1763566 T^{2} - 2048 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−660T+407702T2−660p3T3+p6T4 1 - 660 T + 407702 T^{2} - 660 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−496T+323534T2−496p3T3+p6T4 1 - 496 T + 323534 T^{2} - 496 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−776T+1131046T2−776p3T3+p6T4 1 - 776 T + 1131046 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−532T+1467382T2−532p3T3+p6T4 1 - 532 T + 1467382 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+1220T+586694T2+1220p3T3+p6T4 1 + 1220 T + 586694 T^{2} + 1220 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

−10.04168311112348861698382369727, −9.779448909852847382546060161511, −9.376116940647493922457025597649, −9.152022641719708945015589347488, −8.303183989998309634106053644143, −7.970385320073127502585552743213, −7.39592972643012927178675942850, −7.09610897635939671834875333761, −6.55422001671749528302404139658, −6.35944992436599034656280145784, −5.33154424284670448438037634895, −5.18184927906689454548206041247, −4.93416257826124655930273353864, −4.76127678371367661293449945380, −3.39109107281374371855001263709, −3.37133580911363636842801467903, −2.29645707107620623293386985922, −1.73238364787003067350675067632, −1.25834291926636198745532918757, −0.30873367226111079688215583878, 0.30873367226111079688215583878, 1.25834291926636198745532918757, 1.73238364787003067350675067632, 2.29645707107620623293386985922, 3.37133580911363636842801467903, 3.39109107281374371855001263709, 4.76127678371367661293449945380, 4.93416257826124655930273353864, 5.18184927906689454548206041247, 5.33154424284670448438037634895, 6.35944992436599034656280145784, 6.55422001671749528302404139658, 7.09610897635939671834875333761, 7.39592972643012927178675942850, 7.970385320073127502585552743213, 8.303183989998309634106053644143, 9.152022641719708945015589347488, 9.376116940647493922457025597649, 9.779448909852847382546060161511, 10.04168311112348861698382369727

Graph of the ZZ-function along the critical line