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Properties

Label 2-315-7.2-c1-0-3
Degree $2$
Conductor $315$
Sign $0.0725 - 0.997i$
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  + (−0.207 + 0.358i)2-s + (0.914 + 1.58i)4-s + (0.5 − 0.866i)5-s + (−1.62 + 2.09i)7-s − 1.58·8-s + (0.207 + 0.358i)10-s + (2.41 + 4.18i)11-s + 0.828·13-s + (−0.414 − 1.01i)14-s + (−1.49 + 2.59i)16-s + (−0.414 − 0.717i)17-s + (1.41 − 2.44i)19-s + 1.82·20-s − 2·22-s + (−1.20 + 2.09i)23-s + ⋯
L(s)  = 1  + (−0.146 + 0.253i)2-s + (0.457 + 0.791i)4-s + (0.223 − 0.387i)5-s + (−0.612 + 0.790i)7-s − 0.560·8-s + (0.0654 + 0.113i)10-s + (0.727 + 1.26i)11-s + 0.229·13-s + (−0.110 − 0.271i)14-s + (−0.374 + 0.649i)16-s + (−0.100 − 0.174i)17-s + (0.324 − 0.561i)19-s + 0.408·20-s − 0.426·22-s + (−0.251 + 0.435i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0725 - 0.997i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.936147 + 0.870512i\)
\(L(\frac12)\) \(\approx\) \(0.936147 + 0.870512i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 2.09i)T \)
good2 \( 1 + (0.207 - 0.358i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-2.41 - 4.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.828T + 13T^{2} \)
17 \( 1 + (0.414 + 0.717i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.20 - 2.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.17T + 41T^{2} \)
43 \( 1 - 6.41T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.41 + 5.91i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.24 + 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.74 + 9.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.20 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + (2.41 + 4.18i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.58 - 7.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (-1.32 + 2.30i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−12.18930280001432078109982920716, −11.12948149340809958646328551892, −9.668461009330218093302053697486, −9.130072018662917424012072850376, −8.098377509703029302634306887148, −6.98318635479272987737139550060, −6.25583308427298575003913739300, −4.87181149079669919287181110819, −3.46683187103161839643653168903, −2.13508039717545889927162277023, 1.03163822746840537048396106015, 2.81222274687923773276267466078, 4.04303189579587745515438005750, 5.87454969174582856642392667454, 6.32676375796448102580509397893, 7.45338378554550585379725342124, 8.838141371275603192504486063342, 9.780668625912418662486775924832, 10.55075782806352229108827288507, 11.20875605602377440552672737787

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