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Properties

Label 2-75712-1.1-c1-0-0
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  + 2·5-s − 7-s − 3·9-s − 4·11-s − 6·17-s − 8·23-s − 25-s + 10·29-s − 8·31-s − 2·35-s + 6·37-s + 6·41-s + 4·43-s − 6·45-s − 8·47-s + 49-s − 6·53-s − 8·55-s − 8·59-s − 10·61-s + 3·63-s − 4·67-s − 8·71-s − 2·73-s + 4·77-s − 8·79-s + 9·81-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s − 1.45·17-s − 1.66·23-s − 1/5·25-s + 1.85·29-s − 1.43·31-s − 0.338·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s − 1.04·59-s − 1.28·61-s + 0.377·63-s − 0.488·67-s − 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.900·79-s + 81-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1200865424\)
\(L(\frac12)\) \(\approx\) \(0.1200865424\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.03930775970412, −13.57123524189684, −13.13976578565492, −12.64693167240391, −12.15403726391445, −11.46599108471443, −10.99319519650178, −10.55338994604871, −10.03143930501380, −9.527148811934367, −9.032300289309853, −8.507793812839307, −7.922191178258361, −7.532041428718496, −6.635286625267975, −6.112564247667502, −5.908253385914752, −5.260872843396848, −4.567984678307300, −4.097802447524988, −3.066286276931254, −2.675976776453583, −2.179155879988052, −1.449219187681503, −0.1053190661743869, 0.1053190661743869, 1.449219187681503, 2.179155879988052, 2.675976776453583, 3.066286276931254, 4.097802447524988, 4.567984678307300, 5.260872843396848, 5.908253385914752, 6.112564247667502, 6.635286625267975, 7.532041428718496, 7.922191178258361, 8.507793812839307, 9.032300289309853, 9.527148811934367, 10.03143930501380, 10.55338994604871, 10.99319519650178, 11.46599108471443, 12.15403726391445, 12.64693167240391, 13.13976578565492, 13.57123524189684, 14.03930775970412

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