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Properties

Label 2-312-104.101-c1-0-1
Degree $2$
Conductor $312$
Sign $-0.130 - 0.991i$
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  + (−1.29 − 0.565i)2-s + (−0.866 + 0.5i)3-s + (1.35 + 1.46i)4-s − 1.25·5-s + (1.40 − 0.157i)6-s + (1.81 + 1.05i)7-s + (−0.932 − 2.67i)8-s + (0.499 − 0.866i)9-s + (1.63 + 0.712i)10-s + (−0.586 − 1.01i)11-s + (−1.91 − 0.590i)12-s + (−2.36 + 2.71i)13-s + (−1.76 − 2.39i)14-s + (1.09 − 0.629i)15-s + (−0.303 + 3.98i)16-s + (−2.24 + 3.89i)17-s + ⋯
L(s)  = 1  + (−0.916 − 0.400i)2-s + (−0.499 + 0.288i)3-s + (0.679 + 0.733i)4-s − 0.563·5-s + (0.573 − 0.0644i)6-s + (0.687 + 0.397i)7-s + (−0.329 − 0.944i)8-s + (0.166 − 0.288i)9-s + (0.516 + 0.225i)10-s + (−0.176 − 0.306i)11-s + (−0.551 − 0.170i)12-s + (−0.656 + 0.754i)13-s + (−0.471 − 0.638i)14-s + (0.281 − 0.162i)15-s + (−0.0757 + 0.997i)16-s + (−0.545 + 0.944i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.130 - 0.991i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ -0.130 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312859 + 0.356593i\)
\(L(\frac12)\) \(\approx\) \(0.312859 + 0.356593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.565i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (2.36 - 2.71i)T \)
good5 \( 1 + 1.25T + 5T^{2} \)
7 \( 1 + (-1.81 - 1.05i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.586 + 1.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.24 - 3.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.575 - 0.997i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.69 - 6.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.34 - 1.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.682iT - 31T^{2} \)
37 \( 1 + (-4.79 - 8.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.84 - 3.95i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.32 - 1.34i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.71iT - 47T^{2} \)
53 \( 1 - 9.95iT - 53T^{2} \)
59 \( 1 + (-5.00 + 8.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.93 + 1.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.888 + 1.53i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.05 + 0.609i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 - 2.73T + 79T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + (-4.52 + 2.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.68 + 3.85i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.53146689531108140667074132266, −11.20286647914438959930510984370, −10.09659281362196009681006814669, −9.166961196408779043470527646855, −8.242101126773153487920895750575, −7.37206447442109013916879072126, −6.19880890699837538288485316466, −4.77029756177973393360547361572, −3.49990534640995310905755639530, −1.77333459638525552949400554683, 0.48342683017226808765302048659, 2.37435434413360577972942951154, 4.55455444644598166870390852420, 5.56068849479225768830799658853, 6.94301580719317400549838736978, 7.52431432618487725490271945498, 8.382865180173611766331516690223, 9.543082125466574488968792496148, 10.58804575595807134450603200645, 11.22265111832590305533005168732

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