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Properties

Label 2-1312-41.32-c1-0-30
Degree $2$
Conductor $1312$
Sign $0.331 + 0.943i$
Analytic cond. $10.4763$
Root an. cond. $3.23672$
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  − 2i·5-s + (3 − 3i)7-s + 3i·9-s + (4 − 4i)13-s + (−3 − 3i)17-s + (6 + 6i)19-s − 6·23-s + 25-s + (2 − 2i)29-s − 6·31-s + (−6 − 6i)35-s + 6·37-s + (−4 − 5i)41-s + 6·45-s + (3 + 3i)47-s + ⋯
L(s)  = 1  − 0.894i·5-s + (1.13 − 1.13i)7-s + i·9-s + (1.10 − 1.10i)13-s + (−0.727 − 0.727i)17-s + (1.37 + 1.37i)19-s − 1.25·23-s + 0.200·25-s + (0.371 − 0.371i)29-s − 1.07·31-s + (−1.01 − 1.01i)35-s + 0.986·37-s + (−0.624 − 0.780i)41-s + 0.894·45-s + (0.437 + 0.437i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1312\)    =    \(2^{5} \cdot 41\)
Sign: $0.331 + 0.943i$
Analytic conductor: \(10.4763\)
Root analytic conductor: \(3.23672\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1312} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1312,\ (\ :1/2),\ 0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.924276005\)
\(L(\frac12)\) \(\approx\) \(1.924276005\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 + (4 + 5i)T \)
good3 \( 1 - 3iT^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + (-3 + 3i)T - 7iT^{2} \)
11 \( 1 - 11iT^{2} \)
13 \( 1 + (-4 + 4i)T - 13iT^{2} \)
17 \( 1 + (3 + 3i)T + 17iT^{2} \)
19 \( 1 + (-6 - 6i)T + 19iT^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (-2 + 2i)T - 29iT^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-3 - 3i)T + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 + (-6 - 6i)T + 67iT^{2} \)
71 \( 1 + (-9 + 9i)T - 71iT^{2} \)
73 \( 1 + 12iT - 73T^{2} \)
79 \( 1 + (9 - 9i)T - 79iT^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-5 + 5i)T - 89iT^{2} \)
97 \( 1 + (-5 - 5i)T + 97iT^{2} \)
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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

−9.501418085284460779059325874929, −8.388318984386349664741614546375, −7.927858138553774775359398639272, −7.39857562650960081117665772677, −5.93101537184387399017731578862, −5.16514684409127151400237530347, −4.45618174639019371561735875075, −3.50945493428374834511232491968, −1.85933774204870186897596012079, −0.895367555727343382170894714370, 1.51171402093699181445155323166, 2.59939262525778985380027188086, 3.67525488885812910803106833635, 4.67745359171453829973301412698, 5.80039929502246160298895306554, 6.46383618078303211486933142084, 7.23575387616021305799087228683, 8.379895306690642539680548967067, 8.937800276168340691835853866026, 9.615517958435247271688481508354

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