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Properties

Label 2-975-975.779-c0-0-2
Degree $2$
Conductor $975$
Sign $0.248 + 0.968i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.863 − 0.280i)2-s + (0.587 − 0.809i)3-s + (−0.142 − 0.103i)4-s + (0.987 + 0.156i)5-s + (−0.734 + 0.533i)6-s + (0.627 + 0.863i)8-s + (−0.309 − 0.951i)9-s + (−0.809 − 0.412i)10-s + (−0.0966 + 0.297i)11-s + (−0.166 + 0.0542i)12-s + (0.951 − 0.309i)13-s + (0.707 − 0.707i)15-s + (−0.245 − 0.754i)16-s + 0.907i·18-s + (−0.124 − 0.124i)20-s + ⋯
L(s)  = 1  + (−0.863 − 0.280i)2-s + (0.587 − 0.809i)3-s + (−0.142 − 0.103i)4-s + (0.987 + 0.156i)5-s + (−0.734 + 0.533i)6-s + (0.627 + 0.863i)8-s + (−0.309 − 0.951i)9-s + (−0.809 − 0.412i)10-s + (−0.0966 + 0.297i)11-s + (−0.166 + 0.0542i)12-s + (0.951 − 0.309i)13-s + (0.707 − 0.707i)15-s + (−0.245 − 0.754i)16-s + 0.907i·18-s + (−0.124 − 0.124i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.248 + 0.968i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ 0.248 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8730594017\)
\(L(\frac12)\) \(\approx\) \(0.8730594017\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (-0.987 - 0.156i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
good2 \( 1 + (0.863 + 0.280i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.550 + 1.69i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 + (1.04 - 1.44i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (0.253 + 0.183i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.533 - 0.734i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.437 - 1.34i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−9.701608108072480272508472201378, −9.348410496109021990120676971563, −8.406861197396729364424894436506, −7.86153182164594138201067113954, −6.71121767261732222303970976199, −5.96035712877150655204162892034, −4.89000495199002473330959848076, −3.31401957204235026711755517623, −2.13618708428655400869897223792, −1.27454563877749017762600835090, 1.64223146874534932071114887388, 3.11262136157304470970733054572, 4.14028364794565022191879672368, 5.13632042172916917043007711616, 6.18134253700859526935310597508, 7.21755748737326645978788851991, 8.382498619726031650440277038907, 8.688292081526480033072849009524, 9.458937325539296115617372699363, 10.13703309441723086968754845585

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