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Properties

Label 2-315-45.34-c1-0-7
Degree $2$
Conductor $315$
Sign $0.998 - 0.0458i$
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  + (−0.0813 − 0.0469i)2-s + (−1.71 + 0.268i)3-s + (−0.995 − 1.72i)4-s + (−1.28 + 1.82i)5-s + (0.151 + 0.0585i)6-s + (0.866 + 0.5i)7-s + 0.374i·8-s + (2.85 − 0.919i)9-s + (0.190 − 0.0880i)10-s + (0.322 − 0.558i)11-s + (2.16 + 2.68i)12-s + (4.19 − 2.42i)13-s + (−0.0469 − 0.0813i)14-s + (1.71 − 3.47i)15-s + (−1.97 + 3.41i)16-s + 4.21i·17-s + ⋯
L(s)  = 1  + (−0.0575 − 0.0332i)2-s + (−0.987 + 0.155i)3-s + (−0.497 − 0.862i)4-s + (−0.576 + 0.816i)5-s + (0.0619 + 0.0238i)6-s + (0.327 + 0.188i)7-s + 0.132i·8-s + (0.951 − 0.306i)9-s + (0.0603 − 0.0278i)10-s + (0.0972 − 0.168i)11-s + (0.625 + 0.774i)12-s + (1.16 − 0.672i)13-s + (−0.0125 − 0.0217i)14-s + (0.442 − 0.896i)15-s + (−0.493 + 0.854i)16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.998 - 0.0458i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.998 - 0.0458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.827829 + 0.0190027i\)
\(L(\frac12)\) \(\approx\) \(0.827829 + 0.0190027i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.268i)T \)
5 \( 1 + (1.28 - 1.82i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (0.0813 + 0.0469i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-0.322 + 0.558i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.19 + 2.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.21iT - 17T^{2} \)
19 \( 1 - 7.19T + 19T^{2} \)
23 \( 1 + (-6.44 + 3.72i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.11 - 1.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.253 - 0.439i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.60iT - 37T^{2} \)
41 \( 1 + (-5.46 - 9.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.40 + 3.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.34 - 3.08i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.46iT - 53T^{2} \)
59 \( 1 + (6.91 + 11.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.28 - 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.50 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.74T + 71T^{2} \)
73 \( 1 + 9.89iT - 73T^{2} \)
79 \( 1 + (0.580 - 1.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.43 - 3.13i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.61T + 89T^{2} \)
97 \( 1 + (-6.03 - 3.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.22775392434613389347485483819, −10.92430876974167015335587420067, −10.11219346790819158780900817611, −8.979855936981138330575014391443, −7.78822884115166328669183539869, −6.52506085581686249769918390489, −5.77177544499062593746475068783, −4.74092170927341533704949984114, −3.46637018297279162147497259501, −1.09235389954368669285785332395, 1.01809351653895116458114539204, 3.59464861269649683652989318470, 4.62844537822968343119241081440, 5.44482477066391463580790705457, 7.08576313168167694965881298222, 7.66069491271061527127546924770, 8.875115029226126397937020968820, 9.576831638439476702709662704113, 11.20437955707143593911865452307, 11.63695912064404450243530790695

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