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Properties

Label 2-31e2-31.16-c1-0-14
Degree $2$
Conductor $961$
Sign $-0.996 + 0.0889i$
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  + (−0.309 + 0.224i)2-s + (1.85 + 1.34i)3-s + (−0.572 + 1.76i)4-s − 2.23·5-s − 0.874·6-s + (−0.309 + 0.951i)7-s + (−0.454 − 1.40i)8-s + (0.690 + 2.12i)9-s + (0.690 − 0.502i)10-s + (−1.31 + 4.03i)11-s + (−3.43 + 2.49i)12-s + (5.55 + 4.03i)13-s + (−0.118 − 0.363i)14-s + (−4.13 − 3.00i)15-s + (−2.54 − 1.84i)16-s + (−0.166 − 0.513i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.158i)2-s + (1.06 + 0.776i)3-s + (−0.286 + 0.881i)4-s − 0.999·5-s − 0.356·6-s + (−0.116 + 0.359i)7-s + (−0.160 − 0.495i)8-s + (0.230 + 0.708i)9-s + (0.218 − 0.158i)10-s + (−0.395 + 1.21i)11-s + (−0.990 + 0.719i)12-s + (1.54 + 1.11i)13-s + (−0.0315 − 0.0970i)14-s + (−1.06 − 0.776i)15-s + (−0.636 − 0.462i)16-s + (−0.0404 − 0.124i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.996 + 0.0889i$
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: \((2,\ 961,\ (\ :1/2),\ -0.996 + 0.0889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0504910 - 1.13330i\)
\(L(\frac12)\) \(\approx\) \(0.0504910 - 1.13330i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 - 0.224i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.85 - 1.34i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
7 \( 1 + (0.309 - 0.951i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (1.31 - 4.03i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-5.55 - 4.03i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.166 + 0.513i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.12 + 6.52i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.99 - 2.17i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 + (6.04 - 4.39i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (0.166 - 0.121i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (3 + 2.17i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.62 + 4.98i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.80 - 3.49i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 4.44T + 61T^{2} \)
67 \( 1 - 6T + 67T^{2} \)
71 \( 1 + (-2.30 - 7.10i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.31 - 4.03i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.43 - 10.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.55 + 1.85i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.81 + 5.57i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (2.16 - 6.65i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.16818983922799478845576521975, −9.385652689603866430350590357641, −8.547553691394672277361841467607, −8.363070450120405734665536416138, −7.33858353035506093180951143518, −6.49173487758495986595106999425, −4.71475160994951344202748073129, −4.01164100645533765330520596091, −3.48764152835142104301866320006, −2.26889770184985744718150736994, 0.50238138823720498035102120434, 1.70069504281993286228078324862, 3.22316670130601093682400870099, 3.78523006995041160247245904983, 5.38361719676496592721777031786, 6.16153184137514915843726361027, 7.36398449361435938711918190890, 8.210957972393553052281237864343, 8.451640120865079089093556652711, 9.405051388689390190814140926149

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