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Properties

Label 2-312-8.5-c1-0-20
Degree $2$
Conductor $312$
Sign $0.209 + 0.977i$
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  + (1.17 − 0.791i)2-s + i·3-s + (0.747 − 1.85i)4-s − 3.05i·5-s + (0.791 + 1.17i)6-s − 0.397·7-s + (−0.592 − 2.76i)8-s − 9-s + (−2.41 − 3.57i)10-s − 0.332i·11-s + (1.85 + 0.747i)12-s − i·13-s + (−0.465 + 0.314i)14-s + 3.05·15-s + (−2.88 − 2.77i)16-s + 4.78·17-s + ⋯
L(s)  = 1  + (0.828 − 0.559i)2-s + 0.577i·3-s + (0.373 − 0.927i)4-s − 1.36i·5-s + (0.323 + 0.478i)6-s − 0.150·7-s + (−0.209 − 0.977i)8-s − 0.333·9-s + (−0.764 − 1.13i)10-s − 0.100i·11-s + (0.535 + 0.215i)12-s − 0.277i·13-s + (−0.124 + 0.0840i)14-s + 0.788·15-s + (−0.720 − 0.693i)16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.209 + 0.977i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.209 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54283 - 1.24717i\)
\(L(\frac12)\) \(\approx\) \(1.54283 - 1.24717i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 + 0.791i)T \)
3 \( 1 - iT \)
13 \( 1 + iT \)
good5 \( 1 + 3.05iT - 5T^{2} \)
7 \( 1 + 0.397T + 7T^{2} \)
11 \( 1 + 0.332iT - 11T^{2} \)
17 \( 1 - 4.78T + 17T^{2} \)
19 \( 1 - 8.01iT - 19T^{2} \)
23 \( 1 - 2.73T + 23T^{2} \)
29 \( 1 - 2.88iT - 29T^{2} \)
31 \( 1 - 1.91T + 31T^{2} \)
37 \( 1 + 5.85iT - 37T^{2} \)
41 \( 1 + 2.16T + 41T^{2} \)
43 \( 1 - 8.64iT - 43T^{2} \)
47 \( 1 - 5.45T + 47T^{2} \)
53 \( 1 - 9.73iT - 53T^{2} \)
59 \( 1 + 7.96iT - 59T^{2} \)
61 \( 1 - 8.09iT - 61T^{2} \)
67 \( 1 + 1.70iT - 67T^{2} \)
71 \( 1 - 5.78T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 2.71iT - 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−11.71754391937109721596735325760, −10.52661701442750677861236248592, −9.804218976670707693700617720213, −8.894922125647965014812996452676, −7.74417691682157956611211259085, −5.98966583286604884006077003125, −5.29260002381166089406904732449, −4.30451055191261207854271755435, −3.24299350675899889047533989028, −1.30904694330707654713297634593, 2.52524254936104120793533106496, 3.41324796261238289028414197966, 4.97434984310374478276426533441, 6.22666394250355906548311388296, 6.94167696030423310454187014056, 7.56477138073899444053597097517, 8.796075244453090362105527343908, 10.20835338886076394737523657289, 11.29986630267009524531267392832, 11.85992984791004531882333039992

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