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Properties

Label 2-315-21.5-c1-0-0
Degree $2$
Conductor $315$
Sign $0.557 - 0.830i$
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  + (−0.848 − 0.489i)2-s + (−0.520 − 0.900i)4-s + (−0.5 + 0.866i)5-s + (1.24 + 2.33i)7-s + 2.97i·8-s + (0.848 − 0.489i)10-s + (−4.23 + 2.44i)11-s − 1.19i·13-s + (0.0850 − 2.59i)14-s + (0.418 − 0.725i)16-s + (0.725 + 1.25i)17-s + (6.58 + 3.80i)19-s + 1.04·20-s + 4.79·22-s + (5.42 + 3.13i)23-s + ⋯
L(s)  = 1  + (−0.599 − 0.346i)2-s + (−0.260 − 0.450i)4-s + (−0.223 + 0.387i)5-s + (0.471 + 0.881i)7-s + 1.05i·8-s + (0.268 − 0.154i)10-s + (−1.27 + 0.737i)11-s − 0.331i·13-s + (0.0227 − 0.692i)14-s + (0.104 − 0.181i)16-s + (0.175 + 0.304i)17-s + (1.51 + 0.872i)19-s + 0.232·20-s + 1.02·22-s + (1.13 + 0.653i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.557 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629546 + 0.335594i\)
\(L(\frac12)\) \(\approx\) \(0.629546 + 0.335594i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.24 - 2.33i)T \)
good2 \( 1 + (0.848 + 0.489i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (4.23 - 2.44i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.19iT - 13T^{2} \)
17 \( 1 + (-0.725 - 1.25i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.58 - 3.80i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.42 - 3.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.729iT - 29T^{2} \)
31 \( 1 + (8.44 - 4.87i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.58 - 6.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 - 5.20T + 43T^{2} \)
47 \( 1 + (-4.15 + 7.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.87 + 3.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.08 + 5.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.52 + 2.03i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.28 + 7.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (4.07 - 2.35i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.81 + 8.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.70T + 83T^{2} \)
89 \( 1 + (-0.887 + 1.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.2iT - 97T^{2} \)
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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

−11.62708790043398499348212262507, −10.70552488382636247496911058897, −10.03498704985457689204673656626, −9.090252006459775679472169168771, −8.132153781851851513479318131558, −7.28660293433185701797848473559, −5.52461581289122787736463803092, −5.11733455257594892380586841052, −3.11583926667486190702458578315, −1.73485247185641737065712846508, 0.65794057923429440276648395211, 3.11192984906428994650566435402, 4.39217896790083684082619445851, 5.46739461578227775626493733959, 7.27828072405984919413760918749, 7.53868114216174551921254888903, 8.665382356362993656813740223836, 9.383515125238089319295511709170, 10.58042976036767742201759043106, 11.33833971596800651557260994311

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