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Properties

Label 2-312-104.11-c1-0-18
Degree $2$
Conductor $312$
Sign $0.501 + 0.865i$
Analytic cond. $2.49133$
Root an. cond. $1.57839$
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  + (−1.39 + 0.236i)2-s + (−0.5 − 0.866i)3-s + (1.88 − 0.659i)4-s + (1.86 − 1.86i)5-s + (0.901 + 1.08i)6-s + (4.71 − 1.26i)7-s + (−2.47 + 1.36i)8-s + (−0.499 + 0.866i)9-s + (−2.16 + 3.04i)10-s + (−1.99 − 0.535i)11-s + (−1.51 − 1.30i)12-s + (−3.15 + 1.75i)13-s + (−6.27 + 2.87i)14-s + (−2.54 − 0.683i)15-s + (3.13 − 2.48i)16-s + (3.84 + 2.22i)17-s + ⋯
L(s)  = 1  + (−0.985 + 0.167i)2-s + (−0.288 − 0.499i)3-s + (0.944 − 0.329i)4-s + (0.834 − 0.834i)5-s + (0.368 + 0.444i)6-s + (1.78 − 0.477i)7-s + (−0.875 + 0.482i)8-s + (−0.166 + 0.288i)9-s + (−0.683 + 0.962i)10-s + (−0.602 − 0.161i)11-s + (−0.437 − 0.376i)12-s + (−0.874 + 0.485i)13-s + (−1.67 + 0.768i)14-s + (−0.658 − 0.176i)15-s + (0.782 − 0.622i)16-s + (0.933 + 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.501 + 0.865i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.873313 - 0.503040i\)
\(L(\frac12)\) \(\approx\) \(0.873313 - 0.503040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.236i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (3.15 - 1.75i)T \)
good5 \( 1 + (-1.86 + 1.86i)T - 5iT^{2} \)
7 \( 1 + (-4.71 + 1.26i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.99 + 0.535i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.84 - 2.22i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.62 + 0.703i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.41 + 5.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.143 - 0.0829i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.77 + 3.77i)T - 31iT^{2} \)
37 \( 1 + (1.19 - 4.46i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.06 - 7.71i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (7.03 + 4.05i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.87 + 1.87i)T + 47iT^{2} \)
53 \( 1 + 8.22iT - 53T^{2} \)
59 \( 1 + (-0.352 - 1.31i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.601 - 0.347i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.26 + 4.71i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.46 - 9.19i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (11.2 - 11.2i)T - 73iT^{2} \)
79 \( 1 - 9.91iT - 79T^{2} \)
83 \( 1 + (-3.88 - 3.88i)T + 83iT^{2} \)
89 \( 1 + (-13.1 - 3.52i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.52 - 0.408i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.51807197261609942934935617317, −10.41578858375802369427996569674, −9.734631765959103833956472587952, −8.319055664391217024072695677106, −8.051276313671524721127511649273, −6.87419066079891921933064138317, −5.56282761559221559158403153668, −4.84289030475277231853937895333, −2.17746095141961220483245340732, −1.17649219975015414106203239438, 1.81670692215019467775974361244, 3.00519428276545733137831523625, 5.07311411028045043374379905115, 5.83347494810950093993638082463, 7.37840425944829288059338266574, 7.966575243217084560687004749630, 9.225269165345992233676459435924, 10.14235759507906660518742642693, 10.60526045106149499654835448028, 11.66008086451382607732392078198

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