Properties

Label 2-75712-1.1-c1-0-15
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  + 2·3-s + 5-s + 7-s + 9-s − 4·11-s + 2·15-s − 2·17-s − 19-s + 2·21-s − 7·23-s − 4·25-s − 4·27-s + 5·29-s + 9·31-s − 8·33-s + 35-s − 2·37-s − 2·41-s − 43-s + 45-s − 9·47-s + 49-s − 4·51-s − 3·53-s − 4·55-s − 2·57-s − 14·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.516·15-s − 0.485·17-s − 0.229·19-s + 0.436·21-s − 1.45·23-s − 4/5·25-s − 0.769·27-s + 0.928·29-s + 1.61·31-s − 1.39·33-s + 0.169·35-s − 0.328·37-s − 0.312·41-s − 0.152·43-s + 0.149·45-s − 1.31·47-s + 1/7·49-s − 0.560·51-s − 0.412·53-s − 0.539·55-s − 0.264·57-s − 1.79·61-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

\(L(1)\) \(\approx\) \(2.684480374\)
\(L(\frac12)\) \(\approx\) \(2.684480374\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−14.04991183764400, −13.64762377763932, −13.28674103778157, −12.70829702434335, −12.06573785217273, −11.63375391638355, −10.98238853284892, −10.34497041619030, −10.00996025382702, −9.550682571685487, −8.903702234701465, −8.337855614089340, −7.917398465086785, −7.832575493642742, −6.846664492495329, −6.250314902335821, −5.829858809661196, −4.940104374502246, −4.679253632211253, −3.824528773150964, −3.265124613118343, −2.575112933722164, −2.168367172814718, −1.644549885183759, −0.4600463278814079, 0.4600463278814079, 1.644549885183759, 2.168367172814718, 2.575112933722164, 3.265124613118343, 3.824528773150964, 4.679253632211253, 4.940104374502246, 5.829858809661196, 6.250314902335821, 6.846664492495329, 7.832575493642742, 7.917398465086785, 8.337855614089340, 8.903702234701465, 9.550682571685487, 10.00996025382702, 10.34497041619030, 10.98238853284892, 11.63375391638355, 12.06573785217273, 12.70829702434335, 13.28674103778157, 13.64762377763932, 14.04991183764400

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