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Properties

Label 2-315-315.178-c1-0-5
Degree $2$
Conductor $315$
Sign $-0.0186 - 0.999i$
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  + (−1.48 + 1.48i)2-s + (−1.69 + 0.377i)3-s − 2.42i·4-s + (0.500 − 2.17i)5-s + (1.95 − 3.07i)6-s + (−2.43 + 1.04i)7-s + (0.638 + 0.638i)8-s + (2.71 − 1.27i)9-s + (2.49 + 3.98i)10-s + (0.0441 + 0.0765i)11-s + (0.916 + 4.10i)12-s + (5.14 + 1.37i)13-s + (2.06 − 5.17i)14-s + (−0.0228 + 3.87i)15-s + 2.95·16-s + (−3.40 + 0.913i)17-s + ⋯
L(s)  = 1  + (−1.05 + 1.05i)2-s + (−0.975 + 0.217i)3-s − 1.21i·4-s + (0.223 − 0.974i)5-s + (0.797 − 1.25i)6-s + (−0.918 + 0.395i)7-s + (0.225 + 0.225i)8-s + (0.904 − 0.425i)9-s + (0.790 + 1.26i)10-s + (0.0133 + 0.0230i)11-s + (0.264 + 1.18i)12-s + (1.42 + 0.382i)13-s + (0.550 − 1.38i)14-s + (−0.00590 + 0.999i)15-s + 0.739·16-s + (−0.826 + 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0186 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0186 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0186 - 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.0186 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349873 + 0.356443i\)
\(L(\frac12)\) \(\approx\) \(0.349873 + 0.356443i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 - 0.377i)T \)
5 \( 1 + (-0.500 + 2.17i)T \)
7 \( 1 + (2.43 - 1.04i)T \)
good2 \( 1 + (1.48 - 1.48i)T - 2iT^{2} \)
11 \( 1 + (-0.0441 - 0.0765i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.14 - 1.37i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (3.40 - 0.913i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.57 - 4.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.58 + 0.959i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.609 + 0.351i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.87iT - 31T^{2} \)
37 \( 1 + (-8.44 - 2.26i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.57 - 1.48i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.22 + 0.595i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.06 - 4.06i)T + 47iT^{2} \)
53 \( 1 + (-13.5 + 3.63i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 - 1.36iT - 61T^{2} \)
67 \( 1 + (4.23 - 4.23i)T - 67iT^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 + (0.285 + 1.06i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 - 3.55iT - 79T^{2} \)
83 \( 1 + (0.0856 + 0.319i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-5.71 - 9.89i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.0 + 2.70i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.86881351593524282295181853272, −10.67614940737728757182448340639, −9.705078494895110800698983601670, −9.065331190286820348963315835531, −8.274176538791892536106937574468, −6.88397913984950411410989738317, −6.14955849014601527684822382378, −5.44777298086116606518409586600, −3.92663593764032142892034738668, −1.06406082178464379279299400626, 0.76243119874524599108747863559, 2.52953981490967981258153179579, 3.77593092542693285960236605846, 5.72240727395468573745774437880, 6.64816780340677964897726025749, 7.55253008088109054701606336175, 9.024157868887110392265234833767, 9.814858740638469276711861334211, 10.76185658462418021972720423351, 11.04339141927232577674224025767

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