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Properties

Label 2-637-91.25-c1-0-33
Degree 22
Conductor 637637
Sign −0.932+0.361i-0.932 + 0.361i
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  + (−2.14 + 1.24i)2-s + (0.837 − 1.45i)3-s + (2.07 − 3.59i)4-s + (−0.584 + 0.337i)5-s + 4.15i·6-s + 5.35i·8-s + (0.0969 + 0.167i)9-s + (0.837 − 1.45i)10-s + (−3.88 − 2.24i)11-s + (−3.48 − 6.02i)12-s + (−3.28 − 1.48i)13-s + 1.13i·15-s + (−2.48 − 4.29i)16-s + (−1.64 + 2.84i)17-s + (−0.416 − 0.240i)18-s + (4.52 − 2.60i)19-s + ⋯
L(s)  = 1  + (−1.51 + 0.877i)2-s + (0.483 − 0.837i)3-s + (1.03 − 1.79i)4-s + (−0.261 + 0.150i)5-s + 1.69i·6-s + 1.89i·8-s + (0.0323 + 0.0559i)9-s + (0.264 − 0.458i)10-s + (−1.17 − 0.675i)11-s + (−1.00 − 1.74i)12-s + (−0.911 − 0.410i)13-s + 0.292i·15-s + (−0.620 − 1.07i)16-s + (−0.398 + 0.690i)17-s + (−0.0982 − 0.0567i)18-s + (1.03 − 0.598i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.932+0.361i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.932+0.361i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.932+0.361i-0.932 + 0.361i
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), −0.932+0.361i)(2,\ 637,\ (\ :1/2),\ -0.932 + 0.361i)

Particular Values

L(1)L(1) ≈\approx 0.0156422−0.0835039i0.0156422 - 0.0835039i
L(12)L(\frac12) ≈\approx 0.0156422−0.0835039i0.0156422 - 0.0835039i
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+(3.28+1.48i)T 1 + (3.28 + 1.48i)T
good2 1+(2.14−1.24i)T+(1−1.73i)T2 1 + (2.14 - 1.24i)T + (1 - 1.73i)T^{2}
3 1+(−0.837+1.45i)T+(−1.5−2.59i)T2 1 + (-0.837 + 1.45i)T + (-1.5 - 2.59i)T^{2}
5 1+(0.584−0.337i)T+(2.5−4.33i)T2 1 + (0.584 - 0.337i)T + (2.5 - 4.33i)T^{2}
11 1+(3.88+2.24i)T+(5.5+9.52i)T2 1 + (3.88 + 2.24i)T + (5.5 + 9.52i)T^{2}
17 1+(1.64−2.84i)T+(−8.5−14.7i)T2 1 + (1.64 - 2.84i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.52+2.60i)T+(9.5−16.4i)T2 1 + (-4.52 + 2.60i)T + (9.5 - 16.4i)T^{2}
23 1+(2.38+4.12i)T+(−11.5+19.9i)T2 1 + (2.38 + 4.12i)T + (-11.5 + 19.9i)T^{2}
29 1+9.31T+29T2 1 + 9.31T + 29T^{2}
31 1+(−1.41−0.818i)T+(15.5+26.8i)T2 1 + (-1.41 - 0.818i)T + (15.5 + 26.8i)T^{2}
37 1+(1.25−0.721i)T+(18.5−32.0i)T2 1 + (1.25 - 0.721i)T + (18.5 - 32.0i)T^{2}
41 1−7.92iT−41T2 1 - 7.92iT - 41T^{2}
43 1+4.61T+43T2 1 + 4.61T + 43T^{2}
47 1+(6.81−3.93i)T+(23.5−40.7i)T2 1 + (6.81 - 3.93i)T + (23.5 - 40.7i)T^{2}
53 1+(−1.57+2.73i)T+(−26.5−45.8i)T2 1 + (-1.57 + 2.73i)T + (-26.5 - 45.8i)T^{2}
59 1+(2.20+1.27i)T+(29.5+51.0i)T2 1 + (2.20 + 1.27i)T + (29.5 + 51.0i)T^{2}
61 1+(1.15+2.00i)T+(−30.5+52.8i)T2 1 + (1.15 + 2.00i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.36+3.67i)T+(33.5+58.0i)T2 1 + (6.36 + 3.67i)T + (33.5 + 58.0i)T^{2}
71 1−7.75iT−71T2 1 - 7.75iT - 71T^{2}
73 1+(13.1+7.57i)T+(36.5+63.2i)T2 1 + (13.1 + 7.57i)T + (36.5 + 63.2i)T^{2}
79 1+(7.33+12.6i)T+(−39.5+68.4i)T2 1 + (7.33 + 12.6i)T + (-39.5 + 68.4i)T^{2}
83 1−1.45iT−83T2 1 - 1.45iT - 83T^{2}
89 1+(−6.74+3.89i)T+(44.5−77.0i)T2 1 + (-6.74 + 3.89i)T + (44.5 - 77.0i)T^{2}
97 1−17.9iT−97T2 1 - 17.9iT - 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

−10.01530260007297109347061750372, −9.076342074537408492344748712151, −8.124814170460279770365364209220, −7.74510607348593359507143155454, −7.12168422715009627849052811678, −6.09641517645581063622612280439, −5.05429722371347239303575781157, −2.95402480108783233746895225339, −1.72011818607322419245114619295, −0.06734020104326760646240778092, 1.94581157450727510236722975315, 3.00149781828412601366325325658, 4.07580028615551144219743497939, 5.28953630180208069241433476791, 7.22161761951695681406792931488, 7.67799504509975815709892071645, 8.674806868524324922915613490643, 9.560149101539265594750781248662, 9.853236174982899767436930821717, 10.55291221450846068872417332324

Graph of the ZZ-function along the critical line