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Properties

Label 2-315-9.7-c1-0-8
Degree $2$
Conductor $315$
Sign $0.985 - 0.168i$
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 + (−1.57 + 0.727i)3-s + (−0.913 − 1.58i)4-s + (−0.5 − 0.866i)5-s + (0.305 − 3.37i)6-s + (0.5 − 0.866i)7-s − 0.336·8-s + (1.94 − 2.28i)9-s + 1.95·10-s + (0.582 − 1.00i)11-s + (2.58 + 1.82i)12-s + (−2.22 − 3.85i)13-s + (0.978 + 1.69i)14-s + (1.41 + 0.997i)15-s + (2.15 − 3.73i)16-s − 0.994·17-s + ⋯
L(s)  = 1  + (−0.691 + 1.19i)2-s + (−0.907 + 0.420i)3-s + (−0.456 − 0.791i)4-s + (−0.223 − 0.387i)5-s + (0.124 − 1.37i)6-s + (0.188 − 0.327i)7-s − 0.119·8-s + (0.647 − 0.762i)9-s + 0.618·10-s + (0.175 − 0.304i)11-s + (0.747 + 0.526i)12-s + (−0.616 − 1.06i)13-s + (0.261 + 0.452i)14-s + (0.365 + 0.257i)15-s + (0.539 − 0.934i)16-s − 0.241·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.985 - 0.168i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.985 - 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516432 + 0.0436911i\)
\(L(\frac12)\) \(\approx\) \(0.516432 + 0.0436911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.57 - 0.727i)T \)
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 \( 1 + (-0.582 + 1.00i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.22 + 3.85i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.994T + 17T^{2} \)
19 \( 1 - 7.73T + 19T^{2} \)
23 \( 1 + (-0.300 - 0.520i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.41 - 5.91i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.41 + 5.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.70T + 37T^{2} \)
41 \( 1 + (5.52 + 9.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.55 + 4.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.25 + 10.8i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.43T + 53T^{2} \)
59 \( 1 + (2.49 + 4.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.303 + 0.526i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.67 + 4.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.95T + 71T^{2} \)
73 \( 1 + 9.75T + 73T^{2} \)
79 \( 1 + (-0.188 + 0.326i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.37 + 9.30i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.66T + 89T^{2} \)
97 \( 1 + (8.80 - 15.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.65949858940646883054002134197, −10.56284777905349727755657092692, −9.640162431343989177921750685377, −8.836584616114766249032980220690, −7.59512960155038785078585463067, −7.08240298936026669923202070814, −5.66158255731239503436678816866, −5.22167307175357330807727725331, −3.60382846781582903170302893111, −0.57546271130084498119140163393, 1.40559943128066702048112651663, 2.70013145683981747727626763024, 4.35891019261176764029790611253, 5.72145791832262467407011337474, 6.89811829316646381685720608188, 7.84413064265972162325642470171, 9.264708988993767973897939561399, 9.881731850529861040046049052309, 10.93724896207062405698381514607, 11.65738054004067423692327236231

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