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Properties

Label 2-31e2-31.16-c1-0-60
Degree $2$
Conductor $961$
Sign $-0.997 - 0.0753i$
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  + (1.95 − 1.41i)2-s + (−1.95 − 1.41i)3-s + (1.18 − 3.64i)4-s + 5-s − 5.82·6-s + (0.746 − 2.29i)7-s + (−1.36 − 4.19i)8-s + (0.874 + 2.68i)9-s + (1.95 − 1.41i)10-s + (1.62 − 4.98i)11-s + (−7.47 + 5.43i)12-s + (1.47 + 1.07i)13-s + (−1.80 − 5.54i)14-s + (−1.95 − 1.41i)15-s + (−2.42 − 1.76i)16-s + (0.0530 + 0.163i)17-s + ⋯
L(s)  = 1  + (1.38 − 1.00i)2-s + (−1.12 − 0.819i)3-s + (0.591 − 1.82i)4-s + 0.447·5-s − 2.37·6-s + (0.281 − 0.867i)7-s + (−0.482 − 1.48i)8-s + (0.291 + 0.896i)9-s + (0.617 − 0.448i)10-s + (0.488 − 1.50i)11-s + (−2.15 + 1.56i)12-s + (0.410 + 0.298i)13-s + (−0.481 − 1.48i)14-s + (−0.504 − 0.366i)15-s + (−0.606 − 0.440i)16-s + (0.0128 + 0.0395i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.997 - 0.0753i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ -0.997 - 0.0753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0965924 + 2.55851i\)
\(L(\frac12)\) \(\approx\) \(0.0965924 + 2.55851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 + 1.41i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (1.95 + 1.41i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + (-0.746 + 2.29i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.62 + 4.98i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.47 - 1.07i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.0530 - 0.163i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.28 - 0.932i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.23 - 3.80i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.947 - 0.688i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (7.67 - 5.57i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-7.19 + 5.23i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-1.34 - 0.973i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.0530 + 0.163i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.14 - 5.91i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 + 5.24T + 67T^{2} \)
71 \( 1 + (4.34 + 13.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.18 - 3.64i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.71 - 14.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.29 + 2.39i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.85 + 11.8i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.34 + 10.2i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.20886148213416140093231755693, −8.909320678030390988994398726570, −7.59454033866082650085985133396, −6.50714548773836545000629915671, −5.93956030108375784518281806361, −5.31924054139595423858258647140, −4.14788155802545182707558147771, −3.32752993579582653685878773543, −1.76973640119156840354323532387, −0.926999746360359762691088287768, 2.27064705210076804052635103640, 3.83929580130220073589804698168, 4.64563993681762719192716358557, 5.22981707844418219675641441246, 5.95194659688360011055136543293, 6.57845386758434131250692681364, 7.53788282299152384627974133505, 8.719553289317090767563940607211, 9.734478116487142821882073120487, 10.51670613932175216559625908356

Graph of the $Z$-function along the critical line