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Properties

Label 2-315-315.104-c1-0-23
Degree $2$
Conductor $315$
Sign $0.506 + 0.862i$
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  + (−1.32 + 1.11i)3-s + (1 − 1.73i)4-s + (−1.93 − 1.11i)5-s + (1.32 + 2.29i)7-s + (0.5 − 2.95i)9-s + (−0.311 + 0.179i)11-s + (0.613 + 3.40i)12-s + (3.56 − 6.17i)13-s + (3.81 − 0.686i)15-s + (−1.99 − 3.46i)16-s − 5.75i·17-s + (−3.87 + 2.23i)20-s + (−4.31 − 1.55i)21-s + (2.5 + 4.33i)25-s + (2.64 + 4.47i)27-s + 5.29·28-s + ⋯
L(s)  = 1  + (−0.763 + 0.645i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.499i)5-s + (0.499 + 0.866i)7-s + (0.166 − 0.986i)9-s + (−0.0939 + 0.0542i)11-s + (0.177 + 0.984i)12-s + (0.989 − 1.71i)13-s + (0.984 − 0.177i)15-s + (−0.499 − 0.866i)16-s − 1.39i·17-s + (−0.866 + 0.499i)20-s + (−0.940 − 0.338i)21-s + (0.5 + 0.866i)25-s + (0.509 + 0.860i)27-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.854244 - 0.489171i\)
\(L(\frac12)\) \(\approx\) \(0.854244 - 0.489171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.32 - 1.11i)T \)
5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (-1.32 - 2.29i)T \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
11 \( 1 + (0.311 - 0.179i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.56 + 6.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.75iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.12 + 2.95i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.7 - 6.23i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.3iT - 71T^{2} \)
73 \( 1 - 6.32T + 73T^{2} \)
79 \( 1 + (7.93 + 13.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-15.7 + 9.10i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-7.53 - 13.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.45836874469096189434741058230, −10.76334624809554758106942816636, −9.805001699238757834044378433972, −8.774404623119435671978796823700, −7.71807119135153233741132329192, −6.31365317482727531903050131448, −5.38710147250028367880582974058, −4.75776657264465266181801382952, −3.08052970798113928929772017298, −0.837531570709513230772019113474, 1.74263349474675263328206527857, 3.63013974020717871755136674489, 4.50504657747136154729904136292, 6.45611420907647776319750806204, 6.85460036431125328506662985897, 7.898074439479228279135187339529, 8.508462796826003702563669319041, 10.50048474481568393350834522494, 11.13114707699749195632340551271, 11.67216290739846002100066227897

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