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Properties

Label 2-315-15.8-c1-0-9
Degree $2$
Conductor $315$
Sign $-0.976 + 0.217i$
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  + (−1.45 − 1.45i)2-s + 2.21i·4-s + (1.31 − 1.80i)5-s + (−0.707 + 0.707i)7-s + (0.309 − 0.309i)8-s + (−4.53 + 0.719i)10-s − 5.62i·11-s + (−0.00747 − 0.00747i)13-s + 2.05·14-s + 3.52·16-s + (−1.20 − 1.20i)17-s − 5.69i·19-s + (4.00 + 2.90i)20-s + (−8.17 + 8.17i)22-s + (−4.96 + 4.96i)23-s + ⋯
L(s)  = 1  + (−1.02 − 1.02i)2-s + 1.10i·4-s + (0.587 − 0.809i)5-s + (−0.267 + 0.267i)7-s + (0.109 − 0.109i)8-s + (−1.43 + 0.227i)10-s − 1.69i·11-s + (−0.00207 − 0.00207i)13-s + 0.548·14-s + 0.882·16-s + (−0.293 − 0.293i)17-s − 1.30i·19-s + (0.895 + 0.649i)20-s + (−1.74 + 1.74i)22-s + (−1.03 + 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.976 + 0.217i$
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: \((2,\ 315,\ (\ :1/2),\ -0.976 + 0.217i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0714168 - 0.648395i\)
\(L(\frac12)\) \(\approx\) \(0.0714168 - 0.648395i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 + 1.80i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.45 + 1.45i)T + 2iT^{2} \)
11 \( 1 + 5.62iT - 11T^{2} \)
13 \( 1 + (0.00747 + 0.00747i)T + 13iT^{2} \)
17 \( 1 + (1.20 + 1.20i)T + 17iT^{2} \)
19 \( 1 + 5.69iT - 19T^{2} \)
23 \( 1 + (4.96 - 4.96i)T - 23iT^{2} \)
29 \( 1 + 1.55T + 29T^{2} \)
31 \( 1 + 4.84T + 31T^{2} \)
37 \( 1 + (-3.14 + 3.14i)T - 37iT^{2} \)
41 \( 1 - 9.02iT - 41T^{2} \)
43 \( 1 + (-2.78 - 2.78i)T + 43iT^{2} \)
47 \( 1 + (6.31 + 6.31i)T + 47iT^{2} \)
53 \( 1 + (-4.17 + 4.17i)T - 53iT^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 + (-1.72 + 1.72i)T - 67iT^{2} \)
71 \( 1 + 4.89iT - 71T^{2} \)
73 \( 1 + (-2.69 - 2.69i)T + 73iT^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 + (-5.14 + 5.14i)T - 83iT^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + (-9.91 + 9.91i)T - 97iT^{2} \)
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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

−11.30923988109348494543598451291, −10.18767853186806362061682072202, −9.294079676085590581592935278669, −8.825728164447926381714048647855, −7.910031213829414913677699569946, −6.20046002390503491029810290748, −5.26696561518186426276219456520, −3.47361494193605769557961373381, −2.18928620292958625319639485838, −0.64638391893266245405219369214, 2.05053711979531005055209816864, 3.93209937871713419964549077761, 5.64881562438122910466001022414, 6.59569229166910542189189351733, 7.25208787211168815665035895416, 8.132339455228999663269432949910, 9.370373736111522935742593745840, 10.02209768565894711386966087755, 10.58560457670047139310807191967, 12.15176028727579347747578019691

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