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Properties

Label 2-315-9.4-c1-0-1
Degree $2$
Conductor $315$
Sign $0.0763 - 0.997i$
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  + (0.441 + 0.764i)2-s + (−1.59 + 0.671i)3-s + (0.610 − 1.05i)4-s + (−0.5 + 0.866i)5-s + (−1.21 − 0.924i)6-s + (0.5 + 0.866i)7-s + 2.84·8-s + (2.09 − 2.14i)9-s − 0.882·10-s + (1.91 + 3.31i)11-s + (−0.264 + 2.09i)12-s + (−3.20 + 5.54i)13-s + (−0.441 + 0.764i)14-s + (0.216 − 1.71i)15-s + (0.0340 + 0.0590i)16-s + 7.00·17-s + ⋯
L(s)  = 1  + (0.312 + 0.540i)2-s + (−0.921 + 0.387i)3-s + (0.305 − 0.528i)4-s + (−0.223 + 0.387i)5-s + (−0.497 − 0.377i)6-s + (0.188 + 0.327i)7-s + 1.00·8-s + (0.699 − 0.714i)9-s − 0.279·10-s + (0.577 + 0.999i)11-s + (−0.0763 + 0.605i)12-s + (−0.888 + 1.53i)13-s + (−0.117 + 0.204i)14-s + (0.0559 − 0.443i)15-s + (0.00852 + 0.0147i)16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0763 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0763 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0763 - 0.997i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0763 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.933437 + 0.864664i\)
\(L(\frac12)\) \(\approx\) \(0.933437 + 0.864664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.59 - 0.671i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.441 - 0.764i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-1.91 - 3.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.20 - 5.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.00T + 17T^{2} \)
19 \( 1 + 4.31T + 19T^{2} \)
23 \( 1 + (1.81 - 3.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.54 - 2.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.94 + 5.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.44T + 37T^{2} \)
41 \( 1 + (-2.77 + 4.80i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.96 + 5.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.73 - 3.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.87T + 53T^{2} \)
59 \( 1 + (-1.14 + 1.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.24 + 9.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.27 + 2.20i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.99T + 71T^{2} \)
73 \( 1 - 8.64T + 73T^{2} \)
79 \( 1 + (0.688 + 1.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.09 + 7.09i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.12T + 89T^{2} \)
97 \( 1 + (9.56 + 16.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.88936913202940988908080694841, −11.03227499038968782024565208837, −9.996387263908202373747580838543, −9.462092622046804103064362995829, −7.61266409923041145056487067892, −6.82953189817641156413014142408, −6.03519558217853325877426443089, −4.94288401618545545352870472803, −4.11668047435212955006008311037, −1.82806945629748203288000594548, 1.03839799123368340146022468766, 2.91641075734738881109185083393, 4.24517874522096803338304794797, 5.35471537100822991681312445279, 6.47768674858323234184574035711, 7.75463144948809807695082478726, 8.182143729797579834733794068726, 10.07170988037303128645860075355, 10.70067201002483139878517936629, 11.63969032032046009239930355226

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