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Properties

Label 2-315-63.25-c1-0-5
Degree $2$
Conductor $315$
Sign $-0.513 - 0.858i$
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  − 1.58·2-s + (1.09 + 1.34i)3-s + 0.510·4-s + (−0.5 + 0.866i)5-s + (−1.73 − 2.12i)6-s + (−1.17 − 2.37i)7-s + 2.36·8-s + (−0.609 + 2.93i)9-s + (0.792 − 1.37i)10-s + (1.49 + 2.58i)11-s + (0.557 + 0.685i)12-s + (2.53 + 4.39i)13-s + (1.85 + 3.75i)14-s + (−1.71 + 0.274i)15-s − 4.76·16-s + (−3.18 + 5.51i)17-s + ⋯
L(s)  = 1  − 1.12·2-s + (0.631 + 0.775i)3-s + 0.255·4-s + (−0.223 + 0.387i)5-s + (−0.707 − 0.869i)6-s + (−0.442 − 0.896i)7-s + 0.834·8-s + (−0.203 + 0.979i)9-s + (0.250 − 0.433i)10-s + (0.449 + 0.778i)11-s + (0.161 + 0.197i)12-s + (0.703 + 1.21i)13-s + (0.495 + 1.00i)14-s + (−0.441 + 0.0710i)15-s − 1.19·16-s + (−0.772 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.513 - 0.858i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.513 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321038 + 0.566273i\)
\(L(\frac12)\) \(\approx\) \(0.321038 + 0.566273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.09 - 1.34i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.17 + 2.37i)T \)
good2 \( 1 + 1.58T + 2T^{2} \)
11 \( 1 + (-1.49 - 2.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.53 - 4.39i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.18 - 5.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.69 + 6.40i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.12 - 5.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.418 + 0.725i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.82T + 31T^{2} \)
37 \( 1 + (-2.60 - 4.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.00 + 1.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.06 - 1.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.11T + 47T^{2} \)
53 \( 1 + (1.91 - 3.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 4.78T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 + (-2.01 + 3.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (-5.39 + 9.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.18 + 5.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.444 - 0.770i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.39694903442715549427032084622, −10.73410107459287960530531726966, −9.963152397856426613470047698559, −9.179081903625813966830367443639, −8.505828615618093218786572827737, −7.38903853659771802423274974140, −6.56851528376368062548901428435, −4.40780740325004054369594261969, −3.91751252415406085321863855741, −1.94655800164980277634879856093, 0.64071802180545093583663199915, 2.32594497350695249461695177720, 3.82145408867055633590340363725, 5.67307179130840698661728017676, 6.72618262966100756626912571780, 8.015562068837797469638287977877, 8.519853693665751582855407273312, 9.107041987678399868661374404159, 10.13304120319795437662015591539, 11.28323637427590660241947608108

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