Properties

Label 2-376768-1.1-c1-0-12
Degree $2$
Conductor $376768$
Sign $1$
Analytic cond. $3008.50$
Root an. cond. $54.8498$
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 + 2·13-s − 6·17-s + 8·23-s − 25-s − 4·31-s + 2·35-s + 6·37-s − 6·41-s + 4·43-s − 6·45-s − 4·47-s + 49-s − 6·53-s − 8·55-s + 4·59-s − 14·61-s − 3·63-s + 4·65-s − 4·67-s − 14·73-s − 4·77-s + 16·79-s + 9·81-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.66·23-s − 1/5·25-s − 0.718·31-s + 0.338·35-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.894·45-s − 0.583·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 0.520·59-s − 1.79·61-s − 0.377·63-s + 0.496·65-s − 0.488·67-s − 1.63·73-s − 0.455·77-s + 1.80·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 376768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(376768\)    =    \(2^{6} \cdot 7 \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(3008.50\)
Root analytic conductor: \(54.8498\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 376768,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.236398573\)
\(L(\frac12)\) \(\approx\) \(1.236398573\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
29 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.60656320147963, −11.94772316648360, −11.49515464978938, −10.99400310314412, −10.66602913519293, −10.54971713986259, −9.559153259003941, −9.334564193367867, −8.908790712103477, −8.421781414049674, −7.976140632347539, −7.517395283386485, −6.870594185289672, −6.391321960482400, −5.991865522120798, −5.411764267221138, −5.145665162838792, −4.598505865788083, −4.049426953811045, −3.092918875590845, −2.966418596894725, −2.262198118908480, −1.856815968894540, −1.156216765399246, −0.2812869892976576, 0.2812869892976576, 1.156216765399246, 1.856815968894540, 2.262198118908480, 2.966418596894725, 3.092918875590845, 4.049426953811045, 4.598505865788083, 5.145665162838792, 5.411764267221138, 5.991865522120798, 6.391321960482400, 6.870594185289672, 7.517395283386485, 7.976140632347539, 8.421781414049674, 8.908790712103477, 9.334564193367867, 9.559153259003941, 10.54971713986259, 10.66602913519293, 10.99400310314412, 11.49515464978938, 11.94772316648360, 12.60656320147963

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