Properties

Label 2-315-63.4-c1-0-31
Degree $2$
Conductor $315$
Sign $-0.975 - 0.220i$
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  + (1.15 − 1.99i)2-s − 1.73i·3-s + (−1.65 − 2.86i)4-s − 5-s + (−3.45 − 1.99i)6-s + (−2 + 1.73i)7-s − 2.99·8-s − 2.99·9-s + (−1.15 + 1.99i)10-s + (−4.95 + 2.86i)12-s + (2.80 − 4.85i)13-s + (1.15 + 5.98i)14-s + 1.73i·15-s + (−0.151 + 0.262i)16-s + (0.802 − 1.39i)17-s + (−3.45 + 5.98i)18-s + ⋯
L(s)  = 1  + (0.814 − 1.41i)2-s − 0.999i·3-s + (−0.825 − 1.43i)4-s − 0.447·5-s + (−1.41 − 0.814i)6-s + (−0.755 + 0.654i)7-s − 1.06·8-s − 0.999·9-s + (−0.364 + 0.630i)10-s + (−1.43 + 0.825i)12-s + (0.777 − 1.34i)13-s + (0.307 + 1.59i)14-s + 0.447i·15-s + (−0.0378 + 0.0655i)16-s + (0.194 − 0.337i)17-s + (−0.814 + 1.41i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175046 + 1.56826i\)
\(L(\frac12)\) \(\approx\) \(0.175046 + 1.56826i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 + T \)
7 \( 1 + (2 - 1.73i)T \)
good2 \( 1 + (-1.15 + 1.99i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (-2.80 + 4.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.802 + 1.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.80 + 3.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 9.21T + 23T^{2} \)
29 \( 1 + (-3.10 - 5.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.80 + 3.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.80 + 3.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.10 - 8.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.10 - 5.37i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.80 - 6.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.19 - 3.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.10 + 8.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.10 - 8.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1.80 - 3.12i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.40 + 7.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.105 + 0.182i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.19 + 5.53i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.25171620910759363056626446916, −10.82026691040596235899915630176, −9.431104558308133788887231523918, −8.466011060128980650370718340266, −7.18925942726800943464383533080, −5.97204248609557521547636645421, −4.98192216794837775655273937911, −3.27680129183037498951767200410, −2.73806729320438968491123532350, −0.951761251722919839185561829589, 3.52619168210125590335352926624, 4.11916640215874317645006143390, 5.16056481787002054282791985841, 6.34704718144771768557067336144, 7.00547526319344800056887279316, 8.292984231079648262357457908135, 9.064095121030963457209491207313, 10.27814902311065035064362228313, 11.19799606361581754289856272428, 12.38057847844107785228356473839

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