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Properties

Label 2-31e2-31.5-c1-0-56
Degree $2$
Conductor $961$
Sign $0.654 + 0.755i$
Analytic cond. $7.67362$
Root an. cond. $2.77013$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.49002 - 2.05014i\)
\(L(\frac12)\) \(\approx\) \(4.49002 - 2.05014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line