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Properties

Label 2-31e2-31.26-c0-0-0
Degree $2$
Conductor $961$
Sign $0.920 + 0.390i$
Analytic cond. $0.479601$
Root an. cond. $0.692532$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)14-s − 16-s + (0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + 0.999·35-s + (−0.5 + 0.866i)38-s + (0.5 + 0.866i)40-s + (0.5 + 0.866i)41-s + (0.499 − 0.866i)45-s + ⋯
L(s)  = 1  − 2-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)14-s − 16-s + (0.5 + 0.866i)18-s + (0.5 − 0.866i)19-s + 0.999·35-s + (−0.5 + 0.866i)38-s + (0.5 + 0.866i)40-s + (0.5 + 0.866i)41-s + (0.499 − 0.866i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(0.479601\)
Root analytic conductor: \(0.692532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (522, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :0),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6280972522\)
\(L(\frac12)\) \(\approx\) \(0.6280972522\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - 2T + T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.06794638203864506604602634922, −9.391892427626083233226418013775, −8.657610096550896726863209291967, −7.66991490003053347244434078394, −7.02780761390701433802264862238, −6.14970196429457942999566675395, −4.89155167361197287218858136724, −3.82263509061733196005501090517, −2.56231984565716784452682980561, −1.01023699529904380657190733557, 1.39847065949120171872778332725, 2.42244453427440977825002313666, 4.22593792305898296240089803926, 5.30099653810730400135733864673, 5.66956017083484013551260900284, 7.30276593353686463771207913495, 8.082413507693765219498224826699, 8.756267300312007933973234008222, 9.186211640590049942649645553811, 10.13490038133427654622095782426

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