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Properties

Label 2-315-7.6-c2-0-12
Degree $2$
Conductor $315$
Sign $0.290 + 0.956i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.50·2-s + 8.27·4-s + 2.23i·5-s + (−6.69 + 2.03i)7-s − 14.9·8-s − 7.83i·10-s + 2.03·11-s − 18.0i·13-s + (23.4 − 7.13i)14-s + 19.3·16-s − 1.07i·17-s + 28.7i·19-s + 18.5i·20-s − 7.11·22-s − 24.8·23-s + ⋯
L(s)  = 1  − 1.75·2-s + 2.06·4-s + 0.447i·5-s + (−0.956 + 0.290i)7-s − 1.87·8-s − 0.783i·10-s + 0.184·11-s − 1.38i·13-s + (1.67 − 0.509i)14-s + 1.21·16-s − 0.0631i·17-s + 1.51i·19-s + 0.925i·20-s − 0.323·22-s − 1.08·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.335412 - 0.248603i\)
\(L(\frac12)\) \(\approx\) \(0.335412 - 0.248603i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - 2.23iT \)
7 \( 1 + (6.69 - 2.03i)T \)
good2 \( 1 + 3.50T + 4T^{2} \)
11 \( 1 - 2.03T + 121T^{2} \)
13 \( 1 + 18.0iT - 169T^{2} \)
17 \( 1 + 1.07iT - 289T^{2} \)
19 \( 1 - 28.7iT - 361T^{2} \)
23 \( 1 + 24.8T + 529T^{2} \)
29 \( 1 - 38.4T + 841T^{2} \)
31 \( 1 + 44.0iT - 961T^{2} \)
37 \( 1 - 37.2T + 1.36e3T^{2} \)
41 \( 1 + 49.9iT - 1.68e3T^{2} \)
43 \( 1 + 9.58T + 1.84e3T^{2} \)
47 \( 1 + 55.6iT - 2.20e3T^{2} \)
53 \( 1 + 57.4T + 2.80e3T^{2} \)
59 \( 1 + 101. iT - 3.48e3T^{2} \)
61 \( 1 + 31.9iT - 3.72e3T^{2} \)
67 \( 1 - 95.7T + 4.48e3T^{2} \)
71 \( 1 - 25.8T + 5.04e3T^{2} \)
73 \( 1 + 95.6iT - 5.32e3T^{2} \)
79 \( 1 - 28.1T + 6.24e3T^{2} \)
83 \( 1 - 103. iT - 6.88e3T^{2} \)
89 \( 1 + 29.3iT - 7.92e3T^{2} \)
97 \( 1 + 67.6iT - 9.40e3T^{2} \)
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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

−10.82610644694541540109863334103, −10.00558036961201462757432040366, −9.655725175409374766506454289520, −8.315219993361771782173933779051, −7.80156221412936665749525576492, −6.58541183719361465300334666477, −5.83769790258135753828314060562, −3.49630684938388322266944754967, −2.23438871562153050752757393300, −0.40519449393391429779084308196, 1.12760618796661099724134653782, 2.67819723528612444357738526283, 4.44773705314974248596570162810, 6.38331313911336425338201946039, 6.89899738089159323275156927693, 8.070972606789545075672336064249, 9.045528560180859835241532081211, 9.513482795355436961462099269987, 10.39831762279356724595718252067, 11.38357042865698302793391431214

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