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Properties

Label 2-31e2-31.2-c1-0-33
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  + (−0.284 − 0.206i)2-s + (−2.34 + 1.70i)3-s + (−0.579 − 1.78i)4-s + 2.97·5-s + 1.01·6-s + (−0.334 − 1.02i)7-s + (−0.420 + 1.29i)8-s + (1.65 − 5.10i)9-s + (−0.845 − 0.614i)10-s + (0.751 + 2.31i)11-s + (4.39 + 3.19i)12-s + (2.32 − 1.68i)13-s + (−0.117 + 0.361i)14-s + (−6.95 + 5.05i)15-s + (−2.64 + 1.92i)16-s + (0.563 − 1.73i)17-s + ⋯
L(s)  = 1  + (−0.201 − 0.146i)2-s + (−1.35 + 0.981i)3-s + (−0.289 − 0.892i)4-s + 1.32·5-s + 0.415·6-s + (−0.126 − 0.388i)7-s + (−0.148 + 0.458i)8-s + (0.553 − 1.70i)9-s + (−0.267 − 0.194i)10-s + (0.226 + 0.697i)11-s + (1.26 + 0.921i)12-s + (0.644 − 0.468i)13-s + (−0.0314 + 0.0966i)14-s + (−1.79 + 1.30i)15-s + (−0.662 + 0.481i)16-s + (0.136 − 0.420i)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: $\chi_{961} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.842443 - 0.409984i\)
\(L(\frac12)\) \(\approx\) \(0.842443 - 0.409984i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.284 + 0.206i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (2.34 - 1.70i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 - 2.97T + 5T^{2} \)
7 \( 1 + (0.334 + 1.02i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-0.751 - 2.31i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.32 + 1.68i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.563 + 1.73i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.71 + 1.24i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.136 + 0.420i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.55 + 1.85i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 3.14T + 37T^{2} \)
41 \( 1 + (-5.61 - 4.08i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (6.80 + 4.94i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-6.44 + 4.67i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.53 + 4.73i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.71 + 7.06i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 + 6.43T + 67T^{2} \)
71 \( 1 + (-0.518 + 1.59i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.43 + 13.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.06 + 3.27i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-10.3 - 7.55i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.698 + 2.14i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.05 - 3.24i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.998004439926826150314464038886, −9.570922095358627939377599698357, −8.695383726610505615075241309218, −6.92437689539222711157383046620, −6.17690406523071487907630526283, −5.49773324211934584012974327825, −4.94335496629229931118912105740, −3.90597722073128139366037717875, −2.02581832494605686947444601214, −0.64826411980691904582851545592, 1.19077355757281115036006334698, 2.40643211683382267972212478578, 3.94215648465998504509032339458, 5.35772839935844441427382160669, 5.98096968635674341070334538538, 6.57408955289048792671858513452, 7.42188479629475622569403481281, 8.523193137595281415247079408239, 9.162058718020316934589750534338, 10.23298747421343391571844639278

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