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Properties

Label 2-315-63.25-c1-0-14
Degree $2$
Conductor $315$
Sign $0.204 + 0.978i$
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  − 1.71·2-s + (−0.324 + 1.70i)3-s + 0.955·4-s + (0.5 − 0.866i)5-s + (0.557 − 2.92i)6-s + (−2.52 + 0.774i)7-s + 1.79·8-s + (−2.78 − 1.10i)9-s + (−0.859 + 1.48i)10-s + (−2.30 − 3.98i)11-s + (−0.310 + 1.62i)12-s + (0.944 + 1.63i)13-s + (4.34 − 1.33i)14-s + (1.31 + 1.13i)15-s − 4.99·16-s + (0.371 − 0.643i)17-s + ⋯
L(s)  = 1  − 1.21·2-s + (−0.187 + 0.982i)3-s + 0.477·4-s + (0.223 − 0.387i)5-s + (0.227 − 1.19i)6-s + (−0.956 + 0.292i)7-s + 0.634·8-s + (−0.929 − 0.368i)9-s + (−0.271 + 0.470i)10-s + (−0.693 − 1.20i)11-s + (−0.0895 + 0.469i)12-s + (0.262 + 0.453i)13-s + (1.16 − 0.355i)14-s + (0.338 + 0.292i)15-s − 1.24·16-s + (0.0901 − 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242918 - 0.197421i\)
\(L(\frac12)\) \(\approx\) \(0.242918 - 0.197421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.324 - 1.70i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.52 - 0.774i)T \)
good2 \( 1 + 1.71T + 2T^{2} \)
11 \( 1 + (2.30 + 3.98i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.944 - 1.63i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.371 + 0.643i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.518 + 0.897i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.38 + 4.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.71 + 8.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (2.82 + 4.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.72 + 6.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.26 - 2.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.91T + 47T^{2} \)
53 \( 1 + (3.94 - 6.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.10T + 59T^{2} \)
61 \( 1 - 4.07T + 61T^{2} \)
67 \( 1 + 7.45T + 67T^{2} \)
71 \( 1 + 6.30T + 71T^{2} \)
73 \( 1 + (-3.33 + 5.77i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.21T + 79T^{2} \)
83 \( 1 + (8.45 - 14.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.48 - 14.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.80 + 10.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.98007844993392578365957888367, −10.35043485523807858547798374173, −9.499539900224717584081976805971, −8.846456204733564516450707451882, −8.170566824982355676642299867304, −6.58102973999091870639874649024, −5.55255975758351528870009177444, −4.28853644389938908126036170901, −2.80540329571518002285616833755, −0.35251609164357298677821915635, 1.51056848501857762129490337333, 3.02689524143186059396821927438, 5.05910650960562331158956220374, 6.50958151958350362209975680981, 7.19358316392032847660229858484, 7.987677256672542864451527638659, 9.026001553272681344313929489653, 10.10973617154114047557896063351, 10.52349234514013356051974667726, 11.75894109680952764112394178691

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