# Properties

 Label 2-9576-1.1-c1-0-106 Degree $2$ Conductor $9576$ Sign $-1$ Analytic cond. $76.4647$ Root an. cond. $8.74441$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $1$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2·5-s − 7-s − 2·11-s − 3.12·13-s + 3.12·17-s − 19-s + 7.12·23-s − 25-s − 9.12·29-s − 1.12·31-s − 2·35-s − 0.876·37-s + 8.24·41-s + 4·43-s + 49-s − 5.12·53-s − 4·55-s − 6.24·59-s − 2·61-s − 6.24·65-s − 14.2·67-s + 9.36·71-s − 10·73-s + 2·77-s + 13.1·79-s + 9.12·83-s + 6.24·85-s + ⋯
 L(s)  = 1 + 0.894·5-s − 0.377·7-s − 0.603·11-s − 0.866·13-s + 0.757·17-s − 0.229·19-s + 1.48·23-s − 0.200·25-s − 1.69·29-s − 0.201·31-s − 0.338·35-s − 0.144·37-s + 1.28·41-s + 0.609·43-s + 0.142·49-s − 0.703·53-s − 0.539·55-s − 0.813·59-s − 0.256·61-s − 0.774·65-s − 1.74·67-s + 1.11·71-s − 1.17·73-s + 0.227·77-s + 1.47·79-s + 1.00·83-s + 0.677·85-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$9576$$    =    $$2^{3} \cdot 3^{2} \cdot 7 \cdot 19$$ Sign: $-1$ Analytic conductor: $$76.4647$$ Root analytic conductor: $$8.74441$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 9576,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
7 $$1 + T$$
19 $$1 + T$$
good5 $$1 - 2T + 5T^{2}$$
11 $$1 + 2T + 11T^{2}$$
13 $$1 + 3.12T + 13T^{2}$$
17 $$1 - 3.12T + 17T^{2}$$
23 $$1 - 7.12T + 23T^{2}$$
29 $$1 + 9.12T + 29T^{2}$$
31 $$1 + 1.12T + 31T^{2}$$
37 $$1 + 0.876T + 37T^{2}$$
41 $$1 - 8.24T + 41T^{2}$$
43 $$1 - 4T + 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 + 5.12T + 53T^{2}$$
59 $$1 + 6.24T + 59T^{2}$$
61 $$1 + 2T + 61T^{2}$$
67 $$1 + 14.2T + 67T^{2}$$
71 $$1 - 9.36T + 71T^{2}$$
73 $$1 + 10T + 73T^{2}$$
79 $$1 - 13.1T + 79T^{2}$$
83 $$1 - 9.12T + 83T^{2}$$
89 $$1 - 0.246T + 89T^{2}$$
97 $$1 - 8.24T + 97T^{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

−7.52976781513683755530328151294, −6.62165980050771849222973935862, −5.88257156651207040240560299792, −5.38710787028660319984339311543, −4.73928513783613514319005326355, −3.73972561851802654551471421950, −2.88819445948934469215030515990, −2.26631660693678829317883843248, −1.30198454774280321651165281559, 0, 1.30198454774280321651165281559, 2.26631660693678829317883843248, 2.88819445948934469215030515990, 3.73972561851802654551471421950, 4.73928513783613514319005326355, 5.38710787028660319984339311543, 5.88257156651207040240560299792, 6.62165980050771849222973935862, 7.52976781513683755530328151294