Properties

Label 2-312-13.9-c1-0-3
Degree $2$
Conductor $312$
Sign $0.999 + 0.0256i$
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  + (0.5 − 0.866i)3-s + 3·5-s + (2 + 3.46i)7-s + (−0.499 − 0.866i)9-s + (−2 + 3.46i)11-s + (−2.5 − 2.59i)13-s + (1.5 − 2.59i)15-s + (−1.5 − 2.59i)17-s + (2 + 3.46i)19-s + 3.99·21-s + (4 − 6.92i)23-s + 4·25-s − 0.999·27-s + (2.5 − 4.33i)29-s − 8·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + 1.34·5-s + (0.755 + 1.30i)7-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)11-s + (−0.693 − 0.720i)13-s + (0.387 − 0.670i)15-s + (−0.363 − 0.630i)17-s + (0.458 + 0.794i)19-s + 0.872·21-s + (0.834 − 1.44i)23-s + 0.800·25-s − 0.192·27-s + (0.464 − 0.804i)29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.999 + 0.0256i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.999 + 0.0256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75284 - 0.0224791i\)
\(L(\frac12)\) \(\approx\) \(1.75284 - 0.0224791i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 5T + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2 - 3.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7 - 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.97203891351371068805972167806, −10.58718193668514673289818130812, −9.721404877693394660012466115518, −8.893177391750017321594638484443, −7.915936716857834631976966885766, −6.80747266974240587771543573211, −5.56754534870062268907340746091, −5.00051044922080561458393268027, −2.61987270067235914312200550890, −1.97972979252517803365857623707, 1.65252734159892672478220869023, 3.24299268288688182379717532262, 4.68413657355429970066777703793, 5.53061107605458742488750484563, 6.87345100907401027477054771786, 7.87525592776815281158952278370, 9.109819490646681994508145771921, 9.746521821197915874565769412383, 10.88746238257520542034844996986, 11.13206431393542394912977985172

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