Properties

Label 2-315-15.8-c1-0-6
Degree $2$
Conductor $315$
Sign $-0.349 - 0.937i$
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  + (1.96 + 1.96i)2-s + 5.71i·4-s + (1.65 − 1.50i)5-s + (0.707 − 0.707i)7-s + (−7.29 + 7.29i)8-s + (6.20 + 0.280i)10-s − 0.248i·11-s + (−3.37 − 3.37i)13-s + 2.77·14-s − 17.2·16-s + (1.75 + 1.75i)17-s − 3.91i·19-s + (8.62 + 9.43i)20-s + (0.487 − 0.487i)22-s + (−2.22 + 2.22i)23-s + ⋯
L(s)  = 1  + (1.38 + 1.38i)2-s + 2.85i·4-s + (0.738 − 0.674i)5-s + (0.267 − 0.267i)7-s + (−2.58 + 2.58i)8-s + (1.96 + 0.0886i)10-s − 0.0749i·11-s + (−0.935 − 0.935i)13-s + 0.742·14-s − 4.31·16-s + (0.426 + 0.426i)17-s − 0.898i·19-s + (1.92 + 2.11i)20-s + (0.104 − 0.104i)22-s + (−0.463 + 0.463i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.349 - 0.937i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.349 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55859 + 2.24381i\)
\(L(\frac12)\) \(\approx\) \(1.55859 + 2.24381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.65 + 1.50i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-1.96 - 1.96i)T + 2iT^{2} \)
11 \( 1 + 0.248iT - 11T^{2} \)
13 \( 1 + (3.37 + 3.37i)T + 13iT^{2} \)
17 \( 1 + (-1.75 - 1.75i)T + 17iT^{2} \)
19 \( 1 + 3.91iT - 19T^{2} \)
23 \( 1 + (2.22 - 2.22i)T - 23iT^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
37 \( 1 + (-1.76 + 1.76i)T - 37iT^{2} \)
41 \( 1 - 7.42iT - 41T^{2} \)
43 \( 1 + (0.716 + 0.716i)T + 43iT^{2} \)
47 \( 1 + (-3.34 - 3.34i)T + 47iT^{2} \)
53 \( 1 + (-4.25 + 4.25i)T - 53iT^{2} \)
59 \( 1 - 4.88T + 59T^{2} \)
61 \( 1 - 9.55T + 61T^{2} \)
67 \( 1 + (5.98 - 5.98i)T - 67iT^{2} \)
71 \( 1 - 6.31iT - 71T^{2} \)
73 \( 1 + (10.3 + 10.3i)T + 73iT^{2} \)
79 \( 1 - 11.2iT - 79T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + 1.91T + 89T^{2} \)
97 \( 1 + (-1.07 + 1.07i)T - 97iT^{2} \)
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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

−12.49316784241039269496246096887, −11.46410200934367839004381732344, −9.918771680474788042537607478185, −8.677737967061111437757700365139, −7.86287776195805363553294965222, −6.93779391102343111599622167028, −5.78052081113694880124021110923, −5.16960198500875387800868699064, −4.19517220664973877874455251253, −2.72598295437535964102393127303, 1.82537741977140221309124362915, 2.72323225185120017843992028112, 4.02026810216807406704887642899, 5.15922647110697978720616795637, 6.00310060176396888746648703283, 7.06317463004377795755351039232, 9.127455872387642763229634114731, 10.00545217185730012760765095547, 10.54743699881453884989981485032, 11.67959066492389932227328617456

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