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Properties

Label 2-315-63.16-c1-0-13
Degree $2$
Conductor $315$
Sign $-0.298 - 0.954i$
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  + (0.805 + 1.39i)2-s + (1.04 + 1.38i)3-s + (−0.296 + 0.513i)4-s − 5-s + (−1.09 + 2.56i)6-s + (2.48 + 0.911i)7-s + 2.26·8-s + (−0.832 + 2.88i)9-s + (−0.805 − 1.39i)10-s − 2.16·11-s + (−1.02 + 0.124i)12-s + (−2.21 − 3.83i)13-s + (0.729 + 4.19i)14-s + (−1.04 − 1.38i)15-s + (2.41 + 4.18i)16-s + (−0.752 − 1.30i)17-s + ⋯
L(s)  = 1  + (0.569 + 0.986i)2-s + (0.601 + 0.799i)3-s + (−0.148 + 0.256i)4-s − 0.447·5-s + (−0.445 + 1.04i)6-s + (0.938 + 0.344i)7-s + 0.800·8-s + (−0.277 + 0.960i)9-s + (−0.254 − 0.441i)10-s − 0.653·11-s + (−0.294 + 0.0358i)12-s + (−0.613 − 1.06i)13-s + (0.194 + 1.12i)14-s + (−0.268 − 0.357i)15-s + (0.604 + 1.04i)16-s + (−0.182 − 0.316i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.298 - 0.954i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.298 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26469 + 1.72089i\)
\(L(\frac12)\) \(\approx\) \(1.26469 + 1.72089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.04 - 1.38i)T \)
5 \( 1 + T \)
7 \( 1 + (-2.48 - 0.911i)T \)
good2 \( 1 + (-0.805 - 1.39i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 2.16T + 11T^{2} \)
13 \( 1 + (2.21 + 3.83i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.752 + 1.30i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.165 + 0.286i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.89T + 23T^{2} \)
29 \( 1 + (1.99 - 3.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.87 + 8.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.09 - 1.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.44 + 5.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.16 + 10.6i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.908 - 1.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.32 - 2.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.78 - 6.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.43 - 4.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.61 - 11.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + (5.31 + 9.20i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.55 + 4.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.99 + 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.99 - 5.18i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.11 + 5.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.95354726440844574656852832642, −10.80749415049506222593904686951, −10.20813668202660258211698176739, −8.827992645744516064017838970041, −7.892184497429209798019060455357, −7.41054102344552324379462010244, −5.66558451506474659112401761295, −5.03809634153918541291176160031, −4.07922449792286357847527336913, −2.47732149279625714994076850671, 1.57633362447262430553327146634, 2.66677954901861024906776472195, 3.95643201972230935672976197738, 4.90688737419908448730301636821, 6.71426726863979420757057131468, 7.70711339111322111619783056865, 8.282730816424490119434370915286, 9.666063971934519105125483348307, 10.83966860686863450023244717732, 11.60034327826753717921880877388

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