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Properties

Label 2-312-104.11-c1-0-13
Degree $2$
Conductor $312$
Sign $0.782 - 0.623i$
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  + (−1.25 + 0.648i)2-s + (0.5 + 0.866i)3-s + (1.15 − 1.63i)4-s + (0.513 − 0.513i)5-s + (−1.19 − 0.764i)6-s + (1.43 − 0.383i)7-s + (−0.399 + 2.80i)8-s + (−0.499 + 0.866i)9-s + (−0.312 + 0.977i)10-s + (1.43 + 0.383i)11-s + (1.99 + 0.188i)12-s + (0.967 − 3.47i)13-s + (−1.54 + 1.40i)14-s + (0.700 + 0.187i)15-s + (−1.31 − 3.77i)16-s + (1.08 + 0.628i)17-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)2-s + (0.288 + 0.499i)3-s + (0.579 − 0.815i)4-s + (0.229 − 0.229i)5-s + (−0.485 − 0.311i)6-s + (0.540 − 0.144i)7-s + (−0.141 + 0.989i)8-s + (−0.166 + 0.288i)9-s + (−0.0986 + 0.309i)10-s + (0.431 + 0.115i)11-s + (0.574 + 0.0544i)12-s + (0.268 − 0.963i)13-s + (−0.414 + 0.376i)14-s + (0.180 + 0.0484i)15-s + (−0.328 − 0.944i)16-s + (0.263 + 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.782 - 0.623i$
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: \((2,\ 312,\ (\ :1/2),\ 0.782 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03612 + 0.362223i\)
\(L(\frac12)\) \(\approx\) \(1.03612 + 0.362223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 - 0.648i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.967 + 3.47i)T \)
good5 \( 1 + (-0.513 + 0.513i)T - 5iT^{2} \)
7 \( 1 + (-1.43 + 0.383i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.43 - 0.383i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.08 - 0.628i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.19 + 1.65i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.68 - 2.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.98 - 3.45i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.65 - 1.65i)T - 31iT^{2} \)
37 \( 1 + (-0.944 + 3.52i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.503 + 1.87i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.93 - 4.57i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.96 - 2.96i)T + 47iT^{2} \)
53 \( 1 + 0.513iT - 53T^{2} \)
59 \( 1 + (3.60 + 13.4i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (12.2 + 7.05i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.88 + 7.02i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.979 + 3.65i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.42 - 4.42i)T - 73iT^{2} \)
79 \( 1 - 3.24iT - 79T^{2} \)
83 \( 1 + (2.89 + 2.89i)T + 83iT^{2} \)
89 \( 1 + (2.05 + 0.549i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.09 + 0.293i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.27633161385054154685511517571, −10.81224293651709282925031392305, −9.532265223023147080505988127261, −9.192211516167843138139586049279, −7.942843742223589738641197550148, −7.32315131743573872282170151535, −5.79026848630067647064921172215, −5.02614269796283081785570717976, −3.26485363398119774022837984080, −1.42406287878626566320581594214, 1.38864710163147980958564630682, 2.65760335570390144583779912068, 4.06674770660443090655884790171, 5.92995996784412337073278152854, 7.04396251844087670379281490245, 7.84018871703869960179242876710, 8.865146577119896054589544864397, 9.548753708295886929858281125147, 10.64248472389579510135488634222, 11.67452898365801309008724554244

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