Properties

Label 2-75712-1.1-c1-0-1
Degree $2$
Conductor $75712$
Sign $1$
Analytic cond. $604.563$
Root an. cond. $24.5878$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s − 2·9-s + 4·11-s + 15-s − 6·17-s + 4·19-s + 21-s − 23-s − 4·25-s + 5·27-s + 6·29-s − 6·31-s − 4·33-s + 35-s + 4·37-s − 8·41-s − 4·43-s + 2·45-s − 6·47-s + 49-s + 6·51-s − 4·53-s − 4·55-s − 4·57-s + 3·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.20·11-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.962·27-s + 1.11·29-s − 1.07·31-s − 0.696·33-s + 0.169·35-s + 0.657·37-s − 1.24·41-s − 0.609·43-s + 0.298·45-s − 0.875·47-s + 1/7·49-s + 0.840·51-s − 0.549·53-s − 0.539·55-s − 0.529·57-s + 0.390·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75712\)    =    \(2^{6} \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(604.563\)
Root analytic conductor: \(24.5878\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{75712} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3080072058\)
\(L(\frac12)\) \(\approx\) \(0.3080072058\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.08717371606695, −13.54301176312960, −13.08689181052195, −12.43221607225836, −11.84116260928897, −11.59465789483601, −11.31732129534377, −10.60195871316023, −10.07696455391793, −9.465727586133880, −8.976316063788532, −8.544660447095177, −7.982929349553671, −7.210754832357194, −6.790563004862296, −6.293026621449218, −5.817314380107164, −5.205662638197381, −4.412132111044564, −4.176529253799576, −3.217036685350549, −2.957691498015414, −1.883785230642510, −1.277631723610494, −0.1946797761453815, 0.1946797761453815, 1.277631723610494, 1.883785230642510, 2.957691498015414, 3.217036685350549, 4.176529253799576, 4.412132111044564, 5.205662638197381, 5.817314380107164, 6.293026621449218, 6.790563004862296, 7.210754832357194, 7.982929349553671, 8.544660447095177, 8.976316063788532, 9.465727586133880, 10.07696455391793, 10.60195871316023, 11.31732129534377, 11.59465789483601, 11.84116260928897, 12.43221607225836, 13.08689181052195, 13.54301176312960, 14.08717371606695

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