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Properties

Label 2-315-9.4-c1-0-16
Degree $2$
Conductor $315$
Sign $-0.615 + 0.788i$
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  + (−0.978 − 1.69i)2-s + (1.72 − 0.181i)3-s + (−0.913 + 1.58i)4-s + (0.5 − 0.866i)5-s + (−1.99 − 2.74i)6-s + (0.5 + 0.866i)7-s − 0.338·8-s + (2.93 − 0.623i)9-s − 1.95·10-s + (−1.97 − 3.42i)11-s + (−1.28 + 2.89i)12-s + (1.77 − 3.07i)13-s + (0.978 − 1.69i)14-s + (0.704 − 1.58i)15-s + (2.15 + 3.73i)16-s − 1.95·17-s + ⋯
L(s)  = 1  + (−0.691 − 1.19i)2-s + (0.994 − 0.104i)3-s + (−0.456 + 0.791i)4-s + (0.223 − 0.387i)5-s + (−0.813 − 1.11i)6-s + (0.188 + 0.327i)7-s − 0.119·8-s + (0.978 − 0.207i)9-s − 0.618·10-s + (−0.596 − 1.03i)11-s + (−0.371 + 0.834i)12-s + (0.491 − 0.852i)13-s + (0.261 − 0.452i)14-s + (0.181 − 0.408i)15-s + (0.539 + 0.934i)16-s − 0.474·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.615 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561341 - 1.15092i\)
\(L(\frac12)\) \(\approx\) \(0.561341 - 1.15092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.181i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.978 + 1.69i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (1.97 + 3.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.77 + 3.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.95T + 17T^{2} \)
19 \( 1 - 0.231T + 19T^{2} \)
23 \( 1 + (-3.86 + 6.69i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.88 - 6.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.00 - 5.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.31T + 37T^{2} \)
41 \( 1 + (4.29 - 7.43i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.72 - 6.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.124 + 0.215i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.16T + 53T^{2} \)
59 \( 1 + (0.323 - 0.561i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.83 - 6.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.30 + 5.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.05T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 + (-2.84 - 4.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.40 - 12.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.12T + 89T^{2} \)
97 \( 1 + (3.10 + 5.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.94657072605056949385846709919, −10.53309242006608594000227277357, −9.402934448856839258569989521345, −8.531372532025548152038262131175, −8.286841091688030183854321631498, −6.56943449254395504285686119580, −5.09410324838251972442125888592, −3.39531383060288808263305151044, −2.60971089004785598970714895295, −1.15405195148561930777836847944, 2.13467894961581313023926891479, 3.75970132575257263536267884388, 5.17303407543286575221402117040, 6.65615295091983844837318949755, 7.30013725368230366165916609877, 8.049888556726549836322637731499, 9.094163535218717117011852250848, 9.668242804294296334718450297783, 10.67381294242237312071478009001, 12.00823519732295239787160776596

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