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Properties

Label 2-637-91.25-c1-0-15
Degree 22
Conductor 637637
Sign 0.326−0.945i0.326 - 0.945i
Analytic cond. 5.086475.08647
Root an. cond. 2.255322.25532
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 − i)2-s + (−1 + 1.73i)3-s + (0.999 − 1.73i)4-s + (−0.866 + 0.5i)5-s + 3.99i·6-s + (−0.499 − 0.866i)9-s + (−0.999 + 1.73i)10-s + (−1.73 − i)11-s + (2 + 3.46i)12-s + (2 + 3i)13-s − 1.99i·15-s + (1.99 + 3.46i)16-s + (−3 + 5.19i)17-s + (−1.73 − 0.999i)18-s + (2.59 − 1.5i)19-s + 1.99i·20-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)2-s + (−0.577 + 0.999i)3-s + (0.499 − 0.866i)4-s + (−0.387 + 0.223i)5-s + 1.63i·6-s + (−0.166 − 0.288i)9-s + (−0.316 + 0.547i)10-s + (−0.522 − 0.301i)11-s + (0.577 + 0.999i)12-s + (0.554 + 0.832i)13-s − 0.516i·15-s + (0.499 + 0.866i)16-s + (−0.727 + 1.26i)17-s + (−0.408 − 0.235i)18-s + (0.596 − 0.344i)19-s + 0.447i·20-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.326−0.945i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.326−0.945i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.326−0.945i0.326 - 0.945i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(116,⋅)\chi_{637} (116, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.326−0.945i)(2,\ 637,\ (\ :1/2),\ 0.326 - 0.945i)

Particular Values

L(1)L(1) ≈\approx 1.57986+1.12591i1.57986 + 1.12591i
L(12)L(\frac12) ≈\approx 1.57986+1.12591i1.57986 + 1.12591i
L(32)L(\frac{3}{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}
ppFp(T)F_p(T)
bad7 1 1
13 1+(−2−3i)T 1 + (-2 - 3i)T
good2 1+(−1.73+i)T+(1−1.73i)T2 1 + (-1.73 + i)T + (1 - 1.73i)T^{2}
3 1+(1−1.73i)T+(−1.5−2.59i)T2 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2}
5 1+(0.866−0.5i)T+(2.5−4.33i)T2 1 + (0.866 - 0.5i)T + (2.5 - 4.33i)T^{2}
11 1+(1.73+i)T+(5.5+9.52i)T2 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2}
17 1+(3−5.19i)T+(−8.5−14.7i)T2 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.59+1.5i)T+(9.5−16.4i)T2 1 + (-2.59 + 1.5i)T + (9.5 - 16.4i)T^{2}
23 1+(−1.5−2.59i)T+(−11.5+19.9i)T2 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1+(2.59+1.5i)T+(15.5+26.8i)T2 1 + (2.59 + 1.5i)T + (15.5 + 26.8i)T^{2}
37 1+(5.19−3i)T+(18.5−32.0i)T2 1 + (5.19 - 3i)T + (18.5 - 32.0i)T^{2}
41 1+10iT−41T2 1 + 10iT - 41T^{2}
43 1−T+43T2 1 - T + 43T^{2}
47 1+(−9.52+5.5i)T+(23.5−40.7i)T2 1 + (-9.52 + 5.5i)T + (23.5 - 40.7i)T^{2}
53 1+(−4.5+7.79i)T+(−26.5−45.8i)T2 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2}
59 1+(−6.92−4i)T+(29.5+51.0i)T2 1 + (-6.92 - 4i)T + (29.5 + 51.0i)T^{2}
61 1+(4+6.92i)T+(−30.5+52.8i)T2 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2}
67 1+(−10.3−6i)T+(33.5+58.0i)T2 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2}
71 1+14iT−71T2 1 + 14iT - 71T^{2}
73 1+(−7.79−4.5i)T+(36.5+63.2i)T2 1 + (-7.79 - 4.5i)T + (36.5 + 63.2i)T^{2}
79 1+(−4.5−7.79i)T+(−39.5+68.4i)T2 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2}
83 1+11iT−83T2 1 + 11iT - 83T^{2}
89 1+(4.33−2.5i)T+(44.5−77.0i)T2 1 + (4.33 - 2.5i)T + (44.5 - 77.0i)T^{2}
97 1−9iT−97T2 1 - 9iT - 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.89144454155606032275775545458, −10.45581351208537378794666676615, −9.239479080165778583562246023206, −8.239686173040367941486580570214, −6.90224394134706125888517357654, −5.69026750013145459272667410582, −5.10515318332415301773350018049, −3.99696899306865472544469393537, −3.61123128126634015875098628370, −2.06293278919537627617618987071, 0.78087273595900525212195996870, 2.77490640378173399498801212924, 4.09479204437946895320422608780, 5.10326906207694039456278244770, 5.83981311277659599794892289170, 6.74279055325829995014729498979, 7.37098900893072621059659357283, 8.180912808651004638748588649052, 9.514053715330464077699280559313, 10.68483941730772315508579619049

Graph of the ZZ-function along the critical line