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Properties

Label 2-312-312.155-c1-0-8
Degree $2$
Conductor $312$
Sign $0.771 - 0.636i$
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  + (−1.38 + 0.277i)2-s + (−0.662 − 1.60i)3-s + (1.84 − 0.770i)4-s + 2.38i·5-s + (1.36 + 2.03i)6-s + 2.63·7-s + (−2.34 + 1.58i)8-s + (−2.12 + 2.12i)9-s + (−0.662 − 3.30i)10-s − 0.764·11-s + (−2.45 − 2.44i)12-s + (−1.05 + 3.44i)13-s + (−3.65 + 0.733i)14-s + (3.81 − 1.58i)15-s + (2.81 − 2.84i)16-s + 0.584i·17-s + ⋯
L(s)  = 1  + (−0.980 + 0.196i)2-s + (−0.382 − 0.923i)3-s + (0.922 − 0.385i)4-s + 1.06i·5-s + (0.556 + 0.830i)6-s + 0.997·7-s + (−0.829 + 0.559i)8-s + (−0.707 + 0.707i)9-s + (−0.209 − 1.04i)10-s − 0.230·11-s + (−0.709 − 0.705i)12-s + (−0.291 + 0.956i)13-s + (−0.977 + 0.195i)14-s + (0.985 − 0.408i)15-s + (0.703 − 0.711i)16-s + 0.141i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.771 - 0.636i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.725467 + 0.260749i\)
\(L(\frac12)\) \(\approx\) \(0.725467 + 0.260749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.277i)T \)
3 \( 1 + (0.662 + 1.60i)T \)
13 \( 1 + (1.05 - 3.44i)T \)
good5 \( 1 - 2.38iT - 5T^{2} \)
7 \( 1 - 2.63T + 7T^{2} \)
11 \( 1 + 0.764T + 11T^{2} \)
17 \( 1 - 0.584iT - 17T^{2} \)
19 \( 1 - 4.28iT - 19T^{2} \)
23 \( 1 - 4.39T + 23T^{2} \)
29 \( 1 - 8.65T + 29T^{2} \)
31 \( 1 - 7.09T + 31T^{2} \)
37 \( 1 + 7.63T + 37T^{2} \)
41 \( 1 + 0.139T + 41T^{2} \)
43 \( 1 + 6.17T + 43T^{2} \)
47 \( 1 - 0.889iT - 47T^{2} \)
53 \( 1 + 0.803T + 53T^{2} \)
59 \( 1 - 9.13T + 59T^{2} \)
61 \( 1 - 9.47iT - 61T^{2} \)
67 \( 1 + 11.5iT - 67T^{2} \)
71 \( 1 + 2.56iT - 71T^{2} \)
73 \( 1 + 13.3iT - 73T^{2} \)
79 \( 1 - 10.9iT - 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 5.68T + 89T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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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

−11.63765829256748081293417487998, −10.81758824646073885574107809333, −10.14403837195095033908959691676, −8.643059017947616730331306638260, −7.924145867287121875104275569000, −6.96009110281208562582223451701, −6.40943099384241352644369188476, −5.06610867047269651681081413209, −2.76547481017400767305238500559, −1.55202250477018695200653061632, 0.879323908230066370108654852814, 2.91355869435407512453869783023, 4.65591259973781244953624223623, 5.33734607914513098904793425235, 6.86409767694750080663289881062, 8.353399392242236458123437120157, 8.606682543955821795429992991980, 9.783266040918751205532747587193, 10.51486370729694339314460002035, 11.40717197016257238753681125954

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