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Properties

Label 2-315-63.5-c1-0-10
Degree $2$
Conductor $315$
Sign $0.974 + 0.226i$
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  − 0.692i·2-s + (−1.66 − 0.465i)3-s + 1.51·4-s + (0.5 + 0.866i)5-s + (−0.322 + 1.15i)6-s + (−0.669 + 2.55i)7-s − 2.43i·8-s + (2.56 + 1.55i)9-s + (0.600 − 0.346i)10-s + (1.74 + 1.00i)11-s + (−2.53 − 0.707i)12-s + (3.66 + 2.11i)13-s + (1.77 + 0.463i)14-s + (−0.431 − 1.67i)15-s + 1.35·16-s + (−1.55 − 2.69i)17-s + ⋯
L(s)  = 1  − 0.489i·2-s + (−0.963 − 0.268i)3-s + 0.759·4-s + (0.223 + 0.387i)5-s + (−0.131 + 0.471i)6-s + (−0.253 + 0.967i)7-s − 0.862i·8-s + (0.855 + 0.517i)9-s + (0.189 − 0.109i)10-s + (0.527 + 0.304i)11-s + (−0.732 − 0.204i)12-s + (1.01 + 0.586i)13-s + (0.473 + 0.123i)14-s + (−0.111 − 0.433i)15-s + 0.337·16-s + (−0.376 − 0.652i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.974 + 0.226i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.974 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26202 - 0.144800i\)
\(L(\frac12)\) \(\approx\) \(1.26202 - 0.144800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.66 + 0.465i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.669 - 2.55i)T \)
good2 \( 1 + 0.692iT - 2T^{2} \)
11 \( 1 + (-1.74 - 1.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.66 - 2.11i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.55 + 2.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.112 + 0.0652i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.326 + 0.188i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.55 + 4.35i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.61iT - 31T^{2} \)
37 \( 1 + (-2.94 + 5.09i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.02 - 1.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.79 - 8.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.62T + 47T^{2} \)
53 \( 1 + (8.28 - 4.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 5.36iT - 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 7.48iT - 71T^{2} \)
73 \( 1 + (1.34 - 0.777i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.71T + 79T^{2} \)
83 \( 1 + (0.308 + 0.534i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.725 + 1.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.3 + 6.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.58347378574227894440494729841, −10.99583335356287447692009121313, −10.01420080675887575773344625611, −9.060485199711449716164137014017, −7.52924945890981752248267775733, −6.36272867870957042941743119152, −6.16091498165501051834604236813, −4.52471467389957708573989536974, −2.88389406861596535760049958606, −1.57510437801153906653426735042, 1.25351789627817076170672159764, 3.53683201448118237740778868684, 4.81678134509684196279024213463, 6.10337815193223465793058684719, 6.49622222437412429424366688793, 7.66237790373752945816416292407, 8.785027676429155992501833883797, 10.17052956422588008320401328470, 10.76592852020514235842533712359, 11.52272537908195903367784213030

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