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Properties

Label 2-315-63.4-c1-0-12
Degree $2$
Conductor $315$
Sign $0.197 - 0.980i$
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  + (0.129 − 0.224i)2-s + (−0.458 + 1.67i)3-s + (0.966 + 1.67i)4-s + 5-s + (0.316 + 0.319i)6-s + (2.52 + 0.775i)7-s + 1.02·8-s + (−2.57 − 1.53i)9-s + (0.129 − 0.224i)10-s − 1.45·11-s + (−3.23 + 0.846i)12-s + (−0.192 + 0.333i)13-s + (0.502 − 0.468i)14-s + (−0.458 + 1.67i)15-s + (−1.79 + 3.11i)16-s + (2.42 − 4.20i)17-s + ⋯
L(s)  = 1  + (0.0918 − 0.159i)2-s + (−0.264 + 0.964i)3-s + (0.483 + 0.836i)4-s + 0.447·5-s + (0.129 + 0.130i)6-s + (0.956 + 0.293i)7-s + 0.361·8-s + (−0.859 − 0.510i)9-s + (0.0410 − 0.0711i)10-s − 0.438·11-s + (−0.934 + 0.244i)12-s + (−0.0533 + 0.0923i)13-s + (0.134 − 0.125i)14-s + (−0.118 + 0.431i)15-s + (−0.449 + 0.779i)16-s + (0.588 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.197 - 0.980i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.197 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20089 + 0.983354i\)
\(L(\frac12)\) \(\approx\) \(1.20089 + 0.983354i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.458 - 1.67i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.52 - 0.775i)T \)
good2 \( 1 + (-0.129 + 0.224i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + 1.45T + 11T^{2} \)
13 \( 1 + (0.192 - 0.333i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.42 + 4.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.194 - 0.337i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.95T + 23T^{2} \)
29 \( 1 + (-2.48 - 4.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.69 - 8.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.98 + 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.47 + 6.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.06 + 1.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.31 + 7.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.14 + 3.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.41 + 7.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.87 - 6.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.15 - 3.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.38T + 71T^{2} \)
73 \( 1 + (-5.49 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.839 + 1.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.11 - 10.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.23 + 5.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.759 + 1.31i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.98647003289222776465477454037, −10.86053216164193365020611318496, −10.27049852472552222237348555990, −9.019614992805220713828465911919, −8.211068112261786130723335029886, −7.09820580563805972259281164023, −5.68140036148672983448491103892, −4.80538590359847110914628259935, −3.56090417076188673585005416390, −2.27612078821231403504833876905, 1.29273891684812567378831029950, 2.37881225861349811055803229912, 4.64328352157569635251332163147, 5.81272101432594404177517312965, 6.33001362804112908683120743360, 7.67470894658096069764223654435, 8.189547991705523271478127969568, 9.873921894633015798373153903469, 10.58897817092737843107482993314, 11.50036374307360517201679271469

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