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Properties

Label 2-315-63.16-c1-0-21
Degree $2$
Conductor $315$
Sign $0.896 - 0.442i$
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  + (0.712 + 1.23i)2-s + (1.38 − 1.04i)3-s + (−0.0164 + 0.0284i)4-s + 5-s + (2.27 + 0.962i)6-s + (−0.526 + 2.59i)7-s + 2.80·8-s + (0.820 − 2.88i)9-s + (0.712 + 1.23i)10-s − 3.23·11-s + (0.00700 + 0.0565i)12-s + (−2.89 − 5.02i)13-s + (−3.57 + 1.19i)14-s + (1.38 − 1.04i)15-s + (2.03 + 3.52i)16-s + (1.62 + 2.81i)17-s + ⋯
L(s)  = 1  + (0.504 + 0.873i)2-s + (0.797 − 0.602i)3-s + (−0.00821 + 0.0142i)4-s + 0.447·5-s + (0.928 + 0.392i)6-s + (−0.198 + 0.980i)7-s + 0.991·8-s + (0.273 − 0.961i)9-s + (0.225 + 0.390i)10-s − 0.974·11-s + (0.00202 + 0.0163i)12-s + (−0.804 − 1.39i)13-s + (−0.955 + 0.320i)14-s + (0.356 − 0.269i)15-s + (0.508 + 0.880i)16-s + (0.393 + 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.896 - 0.442i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.896 - 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23604 + 0.522267i\)
\(L(\frac12)\) \(\approx\) \(2.23604 + 0.522267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.38 + 1.04i)T \)
5 \( 1 - T \)
7 \( 1 + (0.526 - 2.59i)T \)
good2 \( 1 + (-0.712 - 1.23i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 + (2.89 + 5.02i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.62 - 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.38 - 4.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.75T + 23T^{2} \)
29 \( 1 + (-2.61 + 4.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.70 - 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 6.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.27 + 2.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.147 + 0.255i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.03 + 8.71i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.20 - 2.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.311 + 0.540i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.44 + 4.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.247 - 0.427i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.06T + 71T^{2} \)
73 \( 1 + (5.79 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.04 - 13.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.961 + 1.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.95 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.37 + 7.57i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.28895854939043471207495742484, −10.50936869691620995864025148067, −9.896632322488316624884035523255, −8.452504419659902028034080379539, −7.912381883760043127563568474594, −6.80602274359713128110964262004, −5.82446989180971761829486320725, −5.11400801466993576525196947770, −3.20900259153390910778727325405, −1.96900840544878174043493522902, 2.09190077243213535167333938088, 3.06381108779983832374554387341, 4.28251451019918774060831473103, 4.98108838024453302784336800068, 7.02123195718152731119008832919, 7.68053216713914629412016563631, 9.105444368179553003429627069326, 9.903888917264308622720378246083, 10.70360034227695749650242614246, 11.40075516755175883497320169821

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