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Properties

Label 2-315-63.41-c1-0-28
Degree $2$
Conductor $315$
Sign $0.709 + 0.704i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (2.31 − 1.33i)2-s + (0.852 + 1.50i)3-s + (2.58 − 4.48i)4-s + (0.5 − 0.866i)5-s + (3.99 + 2.35i)6-s + (−1.06 + 2.42i)7-s − 8.50i·8-s + (−1.54 + 2.56i)9-s − 2.67i·10-s + (−4.13 + 2.38i)11-s + (8.96 + 0.0835i)12-s + (−3.81 − 2.20i)13-s + (0.772 + 7.04i)14-s + (1.73 + 0.0161i)15-s + (−6.22 − 10.7i)16-s + 2.88·17-s + ⋯
L(s)  = 1  + (1.64 − 0.947i)2-s + (0.491 + 0.870i)3-s + (1.29 − 2.24i)4-s + (0.223 − 0.387i)5-s + (1.63 + 0.962i)6-s + (−0.402 + 0.915i)7-s − 3.00i·8-s + (−0.516 + 0.856i)9-s − 0.847i·10-s + (−1.24 + 0.719i)11-s + (2.58 + 0.0241i)12-s + (−1.05 − 0.610i)13-s + (0.206 + 1.88i)14-s + (0.447 + 0.00416i)15-s + (−1.55 − 2.69i)16-s + 0.699·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.709 + 0.704i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.05077 - 1.25766i\)
\(L(\frac12)\) \(\approx\) \(3.05077 - 1.25766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.852 - 1.50i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.06 - 2.42i)T \)
good2 \( 1 + (-2.31 + 1.33i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (4.13 - 2.38i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.81 + 2.20i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.88T + 17T^{2} \)
19 \( 1 + 1.11iT - 19T^{2} \)
23 \( 1 + (-0.0967 - 0.0558i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.32 + 3.65i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.39 - 1.96i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.197T + 37T^{2} \)
41 \( 1 + (-4.21 + 7.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.01 + 1.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.67 - 6.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + (2.59 - 4.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.71 - 2.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.20 - 5.55i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.343iT - 71T^{2} \)
73 \( 1 + 7.87iT - 73T^{2} \)
79 \( 1 + (6.55 + 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.77 - 6.53i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.94T + 89T^{2} \)
97 \( 1 + (-14.8 + 8.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.93436306174254608099011237686, −10.48070955817691083052791263677, −10.15453740534607357188343375421, −9.180145563997395303106353044183, −7.65585065732530538970183693642, −5.90229434246132434279974358163, −5.15690703718806147295923501761, −4.48007086056515887631317166924, −2.93848459487064583069761827167, −2.42906273465957779263068767610, 2.61780923314903695118167884680, 3.43670859166973381743154886372, 4.83750989720612357398297946525, 6.00897847207845270266786115799, 6.82652273508122256715306178570, 7.55191990087400123411807745961, 8.263688240054343452719430391320, 9.988347118049935387551022228536, 11.29832085790634323454565279378, 12.29881267767059164227356812393

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