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Properties

Label 2-64757-1.1-c1-0-2
Degree $2$
Conductor $64757$
Sign $1$
Analytic cond. $517.087$
Root an. cond. $22.7395$
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-s − 4-s − 2·5-s − 7-s − 3·8-s − 3·9-s − 2·10-s + 11-s + 6·13-s − 14-s − 16-s + 2·17-s − 3·18-s + 8·19-s + 2·20-s + 22-s − 25-s + 6·26-s + 28-s + 4·31-s + 5·32-s + 2·34-s + 2·35-s + 3·36-s + 2·37-s + 8·38-s + 6·40-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.377·7-s − 1.06·8-s − 9-s − 0.632·10-s + 0.301·11-s + 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.83·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s + 1.17·26-s + 0.188·28-s + 0.718·31-s + 0.883·32-s + 0.342·34-s + 0.338·35-s + 1/2·36-s + 0.328·37-s + 1.29·38-s + 0.948·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64757 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64757 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64757\)    =    \(7 \cdot 11 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(517.087\)
Root analytic conductor: \(22.7395\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64757,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.653621037\)
\(L(\frac12)\) \(\approx\) \(2.653621037\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
11 \( 1 - T \)
29 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.11471162544168, −13.81479896815004, −13.35353129723097, −12.72052166288819, −12.19704547474294, −11.80304294996916, −11.25204781015006, −11.06510940504220, −10.07097802755535, −9.538895364465644, −9.121396494071260, −8.484132635454385, −8.108480818221393, −7.619022726988947, −6.784986446084161, −6.155121737329416, −5.772369939287782, −5.253639892739997, −4.592205094134559, −3.844774832885585, −3.456155485029713, −3.213916028532060, −2.312396820171225, −0.9765909372887932, −0.6417064513772510, 0.6417064513772510, 0.9765909372887932, 2.312396820171225, 3.213916028532060, 3.456155485029713, 3.844774832885585, 4.592205094134559, 5.253639892739997, 5.772369939287782, 6.155121737329416, 6.784986446084161, 7.619022726988947, 8.108480818221393, 8.484132635454385, 9.121396494071260, 9.538895364465644, 10.07097802755535, 11.06510940504220, 11.25204781015006, 11.80304294996916, 12.19704547474294, 12.72052166288819, 13.35353129723097, 13.81479896815004, 14.11471162544168

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