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Properties

Label 2-31e2-31.5-c1-0-56
Degree 22
Conductor 961961
Sign 0.654+0.755i0.654 + 0.755i
Analytic cond. 7.673627.67362
Root an. cond. 2.770132.77013
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  + 2.61·2-s + (0.437 − 0.756i)3-s + 4.85·4-s + (−1.11 − 1.93i)5-s + (1.14 − 1.98i)6-s + (0.5 − 0.866i)7-s + 7.47·8-s + (1.11 + 1.93i)9-s + (−2.92 − 5.06i)10-s + (−2.12 − 3.67i)11-s + (2.12 − 3.67i)12-s + (1.31 + 2.27i)13-s + (1.30 − 2.26i)14-s − 1.95·15-s + 9.85·16-s + (−1.85 + 3.20i)17-s + ⋯
L(s)  = 1  + 1.85·2-s + (0.252 − 0.437i)3-s + 2.42·4-s + (−0.499 − 0.866i)5-s + (0.467 − 0.809i)6-s + (0.188 − 0.327i)7-s + 2.64·8-s + (0.372 + 0.645i)9-s + (−0.925 − 1.60i)10-s + (−0.639 − 1.10i)11-s + (0.612 − 1.06i)12-s + (0.363 + 0.629i)13-s + (0.349 − 0.605i)14-s − 0.504·15-s + 2.46·16-s + (−0.448 + 0.777i)17-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=((0.654+0.755i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=((0.654+0.755i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: 0.654+0.755i0.654 + 0.755i
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ961(439,⋅)\chi_{961} (439, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 961, ( :1/2), 0.654+0.755i)(2,\ 961,\ (\ :1/2),\ 0.654 + 0.755i)

Particular Values

L(1)L(1) ≈\approx 4.49002−2.05014i4.49002 - 2.05014i
L(12)L(\frac12) ≈\approx 4.49002−2.05014i4.49002 - 2.05014i
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)
bad31 1 1
good2 1−2.61T+2T2 1 - 2.61T + 2T^{2}
3 1+(−0.437+0.756i)T+(−1.5−2.59i)T2 1 + (-0.437 + 0.756i)T + (-1.5 - 2.59i)T^{2}
5 1+(1.11+1.93i)T+(−2.5+4.33i)T2 1 + (1.11 + 1.93i)T + (-2.5 + 4.33i)T^{2}
7 1+(−0.5+0.866i)T+(−3.5−6.06i)T2 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2}
11 1+(2.12+3.67i)T+(−5.5+9.52i)T2 1 + (2.12 + 3.67i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1.31−2.27i)T+(−6.5+11.2i)T2 1 + (-1.31 - 2.27i)T + (-6.5 + 11.2i)T^{2}
17 1+(1.85−3.20i)T+(−8.5−14.7i)T2 1 + (1.85 - 3.20i)T + (-8.5 - 14.7i)T^{2}
19 1+(0.5−0.866i)T+(−9.5−16.4i)T2 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2}
23 1+2.62T+23T2 1 + 2.62T + 23T^{2}
29 1+0.540T+29T2 1 + 0.540T + 29T^{2}
37 1+(2.12−3.67i)T+(−18.5−32.0i)T2 1 + (2.12 - 3.67i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.736−1.27i)T+(−20.5+35.5i)T2 1 + (-0.736 - 1.27i)T + (-20.5 + 35.5i)T^{2}
43 1+(−4.84+8.39i)T+(−21.5−37.2i)T2 1 + (-4.84 + 8.39i)T + (-21.5 - 37.2i)T^{2}
47 1+9.70T+47T2 1 + 9.70T + 47T^{2}
53 1+(−6.86−11.8i)T+(−26.5+45.8i)T2 1 + (-6.86 - 11.8i)T + (-26.5 + 45.8i)T^{2}
59 1+(5.97−10.3i)T+(−29.5−51.0i)T2 1 + (5.97 - 10.3i)T + (-29.5 - 51.0i)T^{2}
61 1−13.9T+61T2 1 - 13.9T + 61T^{2}
67 1+(3+5.19i)T+(−33.5+58.0i)T2 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2}
71 1+(−0.736−1.27i)T+(−35.5+61.4i)T2 1 + (-0.736 - 1.27i)T + (-35.5 + 61.4i)T^{2}
73 1+(2.12+3.67i)T+(−36.5+63.2i)T2 1 + (2.12 + 3.67i)T + (-36.5 + 63.2i)T^{2}
79 1+(−0.810+1.40i)T+(−39.5−68.4i)T2 1 + (-0.810 + 1.40i)T + (-39.5 - 68.4i)T^{2}
83 1+(−1.58−2.73i)T+(−41.5+71.8i)T2 1 + (-1.58 - 2.73i)T + (-41.5 + 71.8i)T^{2}
89 1+15.3T+89T2 1 + 15.3T + 89T^{2}
97 1+7T+97T2 1 + 7T + 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

−10.43479321895908111275620930198, −8.693923852931748528576924969250, −8.041002773323981646165358670821, −7.20349932819183868157519411949, −6.22531802092609790994123553813, −5.40799917324994296926296640665, −4.45616205812294243222859517793, −3.92017055504253646069073535816, −2.66471075011528838648739515231, −1.49297488491669257150666899869, 2.21264936014318618258699664624, 3.10340142104226912934817451413, 3.88430349662771912351977390043, 4.73110586361315207068764110934, 5.54766683206013300992201288687, 6.71477636231847071538529502754, 7.13261677201397136711678479552, 8.159296769280443686902100522808, 9.607226082398068551389076347079, 10.41741326216289341687404247163

Graph of the ZZ-function along the critical line