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Properties

Label 2-315-15.2-c3-0-31
Degree $2$
Conductor $315$
Sign $-0.288 + 0.957i$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.18 − 3.18i)2-s − 12.3i·4-s + (8.71 + 6.99i)5-s + (−4.94 − 4.94i)7-s + (−13.7 − 13.7i)8-s + (50.0 − 5.48i)10-s − 15.9i·11-s + (43.1 − 43.1i)13-s − 31.5·14-s + 10.9·16-s + (18.2 − 18.2i)17-s − 101. i·19-s + (86.1 − 107. i)20-s + (−50.7 − 50.7i)22-s + (−61.4 − 61.4i)23-s + ⋯
L(s)  = 1  + (1.12 − 1.12i)2-s − 1.53i·4-s + (0.779 + 0.625i)5-s + (−0.267 − 0.267i)7-s + (−0.607 − 0.607i)8-s + (1.58 − 0.173i)10-s − 0.436i·11-s + (0.920 − 0.920i)13-s − 0.602·14-s + 0.170·16-s + (0.260 − 0.260i)17-s − 1.22i·19-s + (0.963 − 1.20i)20-s + (−0.491 − 0.491i)22-s + (−0.556 − 0.556i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.288 + 0.957i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ -0.288 + 0.957i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.907099653\)
\(L(\frac12)\) \(\approx\) \(3.907099653\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-8.71 - 6.99i)T \)
7 \( 1 + (4.94 + 4.94i)T \)
good2 \( 1 + (-3.18 + 3.18i)T - 8iT^{2} \)
11 \( 1 + 15.9iT - 1.33e3T^{2} \)
13 \( 1 + (-43.1 + 43.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (-18.2 + 18.2i)T - 4.91e3iT^{2} \)
19 \( 1 + 101. iT - 6.85e3T^{2} \)
23 \( 1 + (61.4 + 61.4i)T + 1.21e4iT^{2} \)
29 \( 1 - 188.T + 2.43e4T^{2} \)
31 \( 1 + 161.T + 2.97e4T^{2} \)
37 \( 1 + (-205. - 205. i)T + 5.06e4iT^{2} \)
41 \( 1 - 314. iT - 6.89e4T^{2} \)
43 \( 1 + (241. - 241. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-140. + 140. i)T - 1.03e5iT^{2} \)
53 \( 1 + (157. + 157. i)T + 1.48e5iT^{2} \)
59 \( 1 - 8.95T + 2.05e5T^{2} \)
61 \( 1 + 840.T + 2.26e5T^{2} \)
67 \( 1 + (-168. - 168. i)T + 3.00e5iT^{2} \)
71 \( 1 - 598. iT - 3.57e5T^{2} \)
73 \( 1 + (392. - 392. i)T - 3.89e5iT^{2} \)
79 \( 1 + 11.0iT - 4.93e5T^{2} \)
83 \( 1 + (-380. - 380. i)T + 5.71e5iT^{2} \)
89 \( 1 - 108.T + 7.04e5T^{2} \)
97 \( 1 + (-1.12e3 - 1.12e3i)T + 9.12e5iT^{2} \)
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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

−10.98770925967486205499661042646, −10.42262467953554094646679568585, −9.547887424281082649215302053500, −8.146858552026098785290777923044, −6.60270998054207994480575520454, −5.78795804594830807873591851109, −4.69278927055025962836190019349, −3.33308115525313541122691182030, −2.66441592937820028255825175785, −1.11150925231740684501448252628, 1.73143198446154429779794709123, 3.64029480951427141566186568202, 4.61961070923211646429881431207, 5.78999260553829525576950387281, 6.19807000340048875243438547978, 7.39952158000608867483731514391, 8.482851740958144981235687384407, 9.439685653008357313444907367228, 10.51450460911201839047633360232, 12.10882377839711146225769379590

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