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Properties

Label 2-31e2-31.16-c1-0-27
Degree $2$
Conductor $961$
Sign $0.617 - 0.786i$
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  + (−1.02 + 0.744i)2-s + (1.20 + 0.872i)3-s + (−0.122 + 0.376i)4-s − 3.80·5-s − 1.88·6-s + (0.676 − 2.08i)7-s + (−0.937 − 2.88i)8-s + (−0.246 − 0.757i)9-s + (3.89 − 2.82i)10-s + (0.293 − 0.904i)11-s + (−0.475 + 0.345i)12-s + (0.136 + 0.0988i)13-s + (0.856 + 2.63i)14-s + (−4.56 − 3.31i)15-s + (2.46 + 1.79i)16-s + (2.03 + 6.25i)17-s + ⋯
L(s)  = 1  + (−0.724 + 0.526i)2-s + (0.693 + 0.503i)3-s + (−0.0611 + 0.188i)4-s − 1.69·5-s − 0.767·6-s + (0.255 − 0.786i)7-s + (−0.331 − 1.02i)8-s + (−0.0820 − 0.252i)9-s + (1.23 − 0.894i)10-s + (0.0885 − 0.272i)11-s + (−0.137 + 0.0996i)12-s + (0.0377 + 0.0274i)13-s + (0.228 + 0.704i)14-s + (−1.17 − 0.856i)15-s + (0.617 + 0.448i)16-s + (0.492 + 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.617 - 0.786i$
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: \((2,\ 961,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.798062 + 0.388386i\)
\(L(\frac12)\) \(\approx\) \(0.798062 + 0.388386i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 - 0.744i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.20 - 0.872i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + 3.80T + 5T^{2} \)
7 \( 1 + (-0.676 + 2.08i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.293 + 0.904i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.136 - 0.0988i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.03 - 6.25i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.932 + 0.677i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.43 - 4.40i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.08 + 0.785i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 3.87T + 37T^{2} \)
41 \( 1 + (0.265 - 0.192i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-7.79 + 5.66i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (4.56 + 3.31i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.26 - 6.97i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.14 - 1.55i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 1.74T + 61T^{2} \)
67 \( 1 - 0.552T + 67T^{2} \)
71 \( 1 + (-0.351 - 1.08i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.44 + 7.53i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (1.40 + 4.31i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.264 + 0.191i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-4.54 + 13.9i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-4.79 + 14.7i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.976344398720720335512120256015, −8.944735131839610463126625140959, −8.467207538720725709446447655664, −7.69857805733414053833531208337, −7.31802989082336942057871229060, −6.13866036739507224565849463926, −4.40414156029034320653364310884, −3.78076676122217844455404358129, −3.27675057956767222150262778378, −0.77322516631649267501931091960, 0.831055298037688781363658655473, 2.35929568720351234994290427836, 3.10540586921718370597757261285, 4.56418356662793646465592353690, 5.38382944988937926272127342172, 6.91414465914120609612848177324, 7.78812025213131558471839352491, 8.260361449855208444470881151084, 8.940859846412022561321977577078, 9.729706134183286837050511588208

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