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Properties

Label 2-315-63.25-c1-0-31
Degree $2$
Conductor $315$
Sign $-0.945 - 0.326i$
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  + 0.831·2-s + (−0.611 − 1.62i)3-s − 1.30·4-s + (−0.5 + 0.866i)5-s + (−0.507 − 1.34i)6-s + (−2.57 + 0.606i)7-s − 2.75·8-s + (−2.25 + 1.98i)9-s + (−0.415 + 0.719i)10-s + (−1.06 − 1.83i)11-s + (0.799 + 2.12i)12-s + (0.552 + 0.956i)13-s + (−2.14 + 0.504i)14-s + (1.70 + 0.281i)15-s + 0.331·16-s + (−3.19 + 5.53i)17-s + ⋯
L(s)  = 1  + 0.587·2-s + (−0.352 − 0.935i)3-s − 0.654·4-s + (−0.223 + 0.387i)5-s + (−0.207 − 0.549i)6-s + (−0.973 + 0.229i)7-s − 0.972·8-s + (−0.751 + 0.660i)9-s + (−0.131 + 0.227i)10-s + (−0.319 − 0.553i)11-s + (0.230 + 0.612i)12-s + (0.153 + 0.265i)13-s + (−0.572 + 0.134i)14-s + (0.441 + 0.0726i)15-s + 0.0829·16-s + (−0.775 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.945 - 0.326i$
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: \((2,\ 315,\ (\ :1/2),\ -0.945 - 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0173505 + 0.103343i\)
\(L(\frac12)\) \(\approx\) \(0.0173505 + 0.103343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.611 + 1.62i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.57 - 0.606i)T \)
good2 \( 1 - 0.831T + 2T^{2} \)
11 \( 1 + (1.06 + 1.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.552 - 0.956i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.19 - 5.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.76 + 4.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.82 + 6.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.160 - 0.278i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 + (-1.76 - 3.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.63 + 9.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.52 - 7.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.35T + 47T^{2} \)
53 \( 1 + (-5.02 + 8.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.51T + 59T^{2} \)
61 \( 1 + 5.97T + 61T^{2} \)
67 \( 1 - 0.240T + 67T^{2} \)
71 \( 1 - 3.28T + 71T^{2} \)
73 \( 1 + (3.39 - 5.87i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.18T + 79T^{2} \)
83 \( 1 + (2.94 - 5.10i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.74 - 4.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.61 - 14.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.26874702300926515574286045986, −10.48959751270968460879948556833, −8.966916205640809721769726060036, −8.431890653160238113474188271931, −6.86935759345551466493222343276, −6.29506732161648292330083516469, −5.22768336717817415255703235295, −3.82314422813488420386668778003, −2.57462564618641762363952094901, −0.06289983167698247527667375200, 3.21114351663055474829369965878, 4.08138858395679322189640645493, 5.08281455466874657059522705508, 5.87980574667819543039837165109, 7.25404392407616309831560978670, 8.789007592614519944473068918512, 9.421954709844770120560322992977, 10.17419954602703477493129681896, 11.32385978413400140824118055397, 12.29364747055753271370686185576

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