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Properties

Label 2-315-15.8-c1-0-8
Degree $2$
Conductor $315$
Sign $0.632 + 0.774i$
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  + (−0.0241 − 0.0241i)2-s − 1.99i·4-s + (2.20 + 0.362i)5-s + (0.707 − 0.707i)7-s + (−0.0963 + 0.0963i)8-s + (−0.0444 − 0.0619i)10-s − 0.390i·11-s + (−2.20 − 2.20i)13-s − 0.0340·14-s − 3.99·16-s + (4.78 + 4.78i)17-s − 5.50i·19-s + (0.724 − 4.41i)20-s + (−0.00941 + 0.00941i)22-s + (2.18 − 2.18i)23-s + ⋯
L(s)  = 1  + (−0.0170 − 0.0170i)2-s − 0.999i·4-s + (0.986 + 0.162i)5-s + (0.267 − 0.267i)7-s + (−0.0340 + 0.0340i)8-s + (−0.0140 − 0.0195i)10-s − 0.117i·11-s + (−0.610 − 0.610i)13-s − 0.00911·14-s − 0.998·16-s + (1.16 + 1.16i)17-s − 1.26i·19-s + (0.161 − 0.986i)20-s + (−0.00200 + 0.00200i)22-s + (0.456 − 0.456i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.632 + 0.774i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.632 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36508 - 0.647853i\)
\(L(\frac12)\) \(\approx\) \(1.36508 - 0.647853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.20 - 0.362i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (0.0241 + 0.0241i)T + 2iT^{2} \)
11 \( 1 + 0.390iT - 11T^{2} \)
13 \( 1 + (2.20 + 2.20i)T + 13iT^{2} \)
17 \( 1 + (-4.78 - 4.78i)T + 17iT^{2} \)
19 \( 1 + 5.50iT - 19T^{2} \)
23 \( 1 + (-2.18 + 2.18i)T - 23iT^{2} \)
29 \( 1 + 7.17T + 29T^{2} \)
31 \( 1 - 5.38T + 31T^{2} \)
37 \( 1 + (2.75 - 2.75i)T - 37iT^{2} \)
41 \( 1 - 2.54iT - 41T^{2} \)
43 \( 1 + (-6.99 - 6.99i)T + 43iT^{2} \)
47 \( 1 + (-0.537 - 0.537i)T + 47iT^{2} \)
53 \( 1 + (5.19 - 5.19i)T - 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 8.12T + 61T^{2} \)
67 \( 1 + (3.76 - 3.76i)T - 67iT^{2} \)
71 \( 1 - 16.0iT - 71T^{2} \)
73 \( 1 + (-7.17 - 7.17i)T + 73iT^{2} \)
79 \( 1 + 7.35iT - 79T^{2} \)
83 \( 1 + (-6.29 + 6.29i)T - 83iT^{2} \)
89 \( 1 + 9.94T + 89T^{2} \)
97 \( 1 + (-2.17 + 2.17i)T - 97iT^{2} \)
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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

−11.20789595639822608022340595416, −10.50086673162694639488422306805, −9.822285323105778339918519214052, −8.960022727403202389287905550517, −7.62109091076386446378187481843, −6.42907852633181631997310199907, −5.61989131685128053190022459410, −4.68572234940373249132593112342, −2.77084142191779376720192859197, −1.29950305776501047781398242588, 2.00193027926436757127455425277, 3.31869733857517332447275657786, 4.78441049217255975764279414097, 5.81075974918501312589715383207, 7.13088187002602808466607761659, 7.907955130757069672156948550533, 9.142254485739069433913088737816, 9.660090861218574720691400782067, 10.91693812503269157744305769282, 12.15590811394463582999157838083

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