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Properties

Label 2-315-15.2-c1-0-1
Degree $2$
Conductor $315$
Sign $-0.519 - 0.854i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.519 - 0.854i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.519 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.425310 + 0.756436i\)
\(L(\frac12)\) \(\approx\) \(0.425310 + 0.756436i\)
\(L(\frac{3}{2})\) 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

−12.18550569335003662297762700034, −11.23169712841315088558478521013, −10.27921305322339575142161534649, −8.744855264759521457467209803524, −8.275232285321665425332276746611, −7.31414264116015851863572714701, −6.31857510806267859455537487737, −4.54345857681961899487521801955, −3.84008012048766561964907122564, −2.41309186144013700810469577513, 0.61889192115712973724395235383, 2.66701522798562291360951640618, 4.48356380950986879254582257169, 5.04545528529579675597653873188, 6.58844083254614565951192750647, 7.39162007727204939665658431346, 8.555454842053968391562517804941, 9.516918032637647013690234005019, 10.51858963416236187232686332157, 11.32902688345640505298467497298

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