Properties

Label 2-75712-1.1-c1-0-44
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·5-s − 7-s + 9-s − 6·15-s − 6·17-s − 7·19-s − 2·21-s + 3·23-s + 4·25-s − 4·27-s + 9·29-s − 5·31-s + 3·35-s + 2·37-s + 6·41-s + 43-s − 3·45-s − 3·47-s + 49-s − 12·51-s + 9·53-s − 14·57-s + 10·61-s − 63-s + 14·67-s + 6·69-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.54·15-s − 1.45·17-s − 1.60·19-s − 0.436·21-s + 0.625·23-s + 4/5·25-s − 0.769·27-s + 1.67·29-s − 0.898·31-s + 0.507·35-s + 0.328·37-s + 0.937·41-s + 0.152·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s − 1.68·51-s + 1.23·53-s − 1.85·57-s + 1.28·61-s − 0.125·63-s + 1.71·67-s + 0.722·69-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: \(1\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
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

−14.46069893291002, −13.81724938560813, −13.30530770495558, −12.73270197496777, −12.60064403133800, −11.72073469047742, −11.30642345315351, −10.88909646683897, −10.31738926907632, −9.635526833195875, −9.027284761844908, −8.573582696545472, −8.387994701291701, −7.804705055571067, −7.141857210720176, −6.756715833164526, −6.214034663445483, −5.331501438216694, −4.565279541759626, −4.020335171680854, −3.845420489682066, −2.931828490214388, −2.546041554363367, −1.940002268647394, −0.7611642337130772, 0, 0.7611642337130772, 1.940002268647394, 2.546041554363367, 2.931828490214388, 3.845420489682066, 4.020335171680854, 4.565279541759626, 5.331501438216694, 6.214034663445483, 6.756715833164526, 7.141857210720176, 7.804705055571067, 8.387994701291701, 8.573582696545472, 9.027284761844908, 9.635526833195875, 10.31738926907632, 10.88909646683897, 11.30642345315351, 11.72073469047742, 12.60064403133800, 12.73270197496777, 13.30530770495558, 13.81724938560813, 14.46069893291002

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