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Properties

Label 2-315-35.12-c1-0-6
Degree $2$
Conductor $315$
Sign $0.601 - 0.798i$
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  + (0.394 − 0.105i)2-s + (−1.58 + 0.916i)4-s + (2.18 − 0.456i)5-s + (−0.605 + 2.57i)7-s + (−1.10 + 1.10i)8-s + (0.815 − 0.411i)10-s + (0.463 + 0.803i)11-s + (4.08 + 4.08i)13-s + (0.0332 + 1.08i)14-s + (1.51 − 2.62i)16-s + (0.719 + 0.192i)17-s + (−1.21 + 2.11i)19-s + (−3.05 + 2.73i)20-s + (0.267 + 0.267i)22-s + (−1.34 − 5.00i)23-s + ⋯
L(s)  = 1  + (0.278 − 0.0747i)2-s + (−0.793 + 0.458i)4-s + (0.978 − 0.204i)5-s + (−0.228 + 0.973i)7-s + (−0.391 + 0.391i)8-s + (0.257 − 0.130i)10-s + (0.139 + 0.242i)11-s + (1.13 + 1.13i)13-s + (0.00889 + 0.288i)14-s + (0.378 − 0.655i)16-s + (0.174 + 0.0467i)17-s + (−0.279 + 0.484i)19-s + (−0.683 + 0.610i)20-s + (0.0571 + 0.0571i)22-s + (−0.279 − 1.04i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.601 - 0.798i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.601 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27624 + 0.636331i\)
\(L(\frac12)\) \(\approx\) \(1.27624 + 0.636331i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 + 0.456i)T \)
7 \( 1 + (0.605 - 2.57i)T \)
good2 \( 1 + (-0.394 + 0.105i)T + (1.73 - i)T^{2} \)
11 \( 1 + (-0.463 - 0.803i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.08 - 4.08i)T + 13iT^{2} \)
17 \( 1 + (-0.719 - 0.192i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.21 - 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.34 + 5.00i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 8.08iT - 29T^{2} \)
31 \( 1 + (1.05 - 0.607i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.76 + 0.472i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.97iT - 41T^{2} \)
43 \( 1 + (0.781 - 0.781i)T - 43iT^{2} \)
47 \( 1 + (2.70 + 10.0i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.42 - 1.72i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.91 + 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.72 + 2.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.80 + 10.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.89T + 71T^{2} \)
73 \( 1 + (-1.07 + 4.02i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-7.02 - 4.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.91 - 5.91i)T + 83iT^{2} \)
89 \( 1 + (7.78 - 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.89 + 4.89i)T - 97iT^{2} \)
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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

−12.19349819152773880672631178057, −10.88556094683920919426788813136, −9.698923238637604726902546707665, −8.918121774298177407900247285830, −8.443138672243765039720339282770, −6.69225004909048078269007723850, −5.77909906575306898429540725856, −4.78295418485791853996726994929, −3.50578279506636933524966117537, −1.99134616854738652962146700996, 1.09238923881500522381955949816, 3.23261643226638944058324141313, 4.40183043686352551956900296336, 5.72441403871863362705444579737, 6.26336176953195447423247519685, 7.69373380607340667547159784058, 8.859739699149034093311239368147, 9.824768976786652826056258231493, 10.37959586635727304077230761986, 11.33366871052114101707747746105

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