Properties

Label 2-315-63.47-c1-0-19
Degree $2$
Conductor $315$
Sign $0.161 + 0.986i$
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  + (−1.91 + 1.10i)2-s + (0.302 − 1.70i)3-s + (1.45 − 2.51i)4-s − 5-s + (1.30 + 3.60i)6-s + (2.53 − 0.756i)7-s + 2.00i·8-s + (−2.81 − 1.03i)9-s + (1.91 − 1.10i)10-s + 1.58i·11-s + (−3.85 − 3.23i)12-s + (1.44 − 0.832i)13-s + (−4.02 + 4.25i)14-s + (−0.302 + 1.70i)15-s + (0.686 + 1.18i)16-s + (−1.59 − 2.76i)17-s + ⋯
L(s)  = 1  + (−1.35 + 0.782i)2-s + (0.174 − 0.984i)3-s + (0.726 − 1.25i)4-s − 0.447·5-s + (0.534 + 1.47i)6-s + (0.958 − 0.286i)7-s + 0.708i·8-s + (−0.939 − 0.343i)9-s + (0.606 − 0.350i)10-s + 0.479i·11-s + (−1.11 − 0.934i)12-s + (0.400 − 0.231i)13-s + (−1.07 + 1.13i)14-s + (−0.0780 + 0.440i)15-s + (0.171 + 0.297i)16-s + (−0.386 − 0.670i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.161 + 0.986i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.161 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.426450 - 0.362516i\)
\(L(\frac12)\) \(\approx\) \(0.426450 - 0.362516i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.302 + 1.70i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.53 + 0.756i)T \)
good2 \( 1 + (1.91 - 1.10i)T + (1 - 1.73i)T^{2} \)
11 \( 1 - 1.58iT - 11T^{2} \)
13 \( 1 + (-1.44 + 0.832i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.59 + 2.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.82 + 3.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.51iT - 23T^{2} \)
29 \( 1 + (-3.79 - 2.19i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.71 + 2.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.21 + 9.03i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.207 - 0.358i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.14 + 3.72i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.73 - 2.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.301 - 0.174i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.25 + 2.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.88 - 5.70i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.42 - 9.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.45iT - 71T^{2} \)
73 \( 1 + (9.85 - 5.68i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.52 - 6.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.33 + 7.51i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.218 + 0.378i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.10 - 1.79i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.13071389604179166164780883176, −10.56631730560435456394044076095, −8.984331692825566626841052067176, −8.568029125520705884440115709266, −7.61617927593041471252422359645, −7.04912969200634531434806569070, −6.07849919003378064891896974937, −4.44294503310841507223154155516, −2.23369179515868535310328560341, −0.62380099263006709496059919332, 1.78291517481795459114218829834, 3.30185775625281871431394067385, 4.50839354482743839658485332168, 5.91320239095355864134952391447, 7.78721177587159920551139026068, 8.419261658912122656400821215485, 9.028112935918042927766539989145, 10.07043683630638723223696419011, 10.92216417280733100261113678436, 11.32567114222624521618315754266

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