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Properties

Label 2-31e2-31.16-c1-0-45
Degree $2$
Conductor $961$
Sign $0.640 + 0.768i$
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  + (0.557 − 0.405i)2-s + (−0.730 − 0.530i)3-s + (−0.471 + 1.44i)4-s + 3.70·5-s − 0.622·6-s + (0.235 − 0.726i)7-s + (0.750 + 2.31i)8-s + (−0.675 − 2.07i)9-s + (2.06 − 1.50i)10-s + (1.27 − 3.91i)11-s + (1.11 − 0.808i)12-s + (−2.35 − 1.71i)13-s + (−0.162 − 0.500i)14-s + (−2.70 − 1.96i)15-s + (−1.11 − 0.807i)16-s + (−0.404 − 1.24i)17-s + ⋯
L(s)  = 1  + (0.394 − 0.286i)2-s + (−0.421 − 0.306i)3-s + (−0.235 + 0.724i)4-s + 1.65·5-s − 0.254·6-s + (0.0891 − 0.274i)7-s + (0.265 + 0.817i)8-s + (−0.225 − 0.692i)9-s + (0.653 − 0.475i)10-s + (0.383 − 1.18i)11-s + (0.321 − 0.233i)12-s + (−0.652 − 0.474i)13-s + (−0.0434 − 0.133i)14-s + (−0.698 − 0.507i)15-s + (−0.277 − 0.201i)16-s + (−0.0980 − 0.301i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.640 + 0.768i$
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: \((2,\ 961,\ (\ :1/2),\ 0.640 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91843 - 0.898712i\)
\(L(\frac12)\) \(\approx\) \(1.91843 - 0.898712i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.557 + 0.405i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.730 + 0.530i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 - 3.70T + 5T^{2} \)
7 \( 1 + (-0.235 + 0.726i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-1.27 + 3.91i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.35 + 1.71i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.404 + 1.24i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.07 + 2.23i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.01 - 3.12i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.96 + 2.87i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + (0.611 - 0.444i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (5.79 - 4.20i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-0.708 - 0.514i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.10 + 3.40i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.750 + 0.545i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 + 2.08T + 67T^{2} \)
71 \( 1 + (2.39 + 7.35i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.74 + 5.38i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.35 - 13.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (11.4 - 8.29i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-1.37 + 4.21i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.63 - 5.01i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.741847123430036486277625761267, −9.272050662663983859327651962561, −8.318151916744360555789089541468, −7.24711334476776333165193051262, −6.27125763611269876321499780774, −5.61937509295240985994172197626, −4.74331050728883826313471632315, −3.33144063043049574747778048643, −2.57467426820412527803923390665, −1.01526227929697719084339838003, 1.55411652555355709711813733558, 2.43686632768528485869138414973, 4.42517810937677609749124025724, 5.01681404084793411080954168572, 5.73098388928824096545678459546, 6.42337488557881401457034175082, 7.29077025566269957957036362090, 8.765107445991092254123262211834, 9.608998159764387312972521699179, 10.06033374139550923725861898938

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