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Properties

Label 2-83790-1.1-c1-0-133
Degree $2$
Conductor $83790$
Sign $-1$
Analytic cond. $669.066$
Root an. cond. $25.8663$
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-s + 4-s + 5-s − 8-s − 10-s + 4·11-s + 2·13-s + 16-s − 2·17-s + 19-s + 20-s − 4·22-s + 8·23-s + 25-s − 2·26-s − 6·29-s − 4·31-s − 32-s + 2·34-s − 10·37-s − 38-s − 40-s − 2·41-s + 12·43-s + 4·44-s − 8·46-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 0.229·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.392·26-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s − 1.64·37-s − 0.162·38-s − 0.158·40-s − 0.312·41-s + 1.82·43-s + 0.603·44-s − 1.17·46-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(83790\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(669.066\)
Root analytic conductor: \(25.8663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 83790,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
19 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.12392894809639, −13.84382797770955, −13.09926284988156, −12.70166004256266, −12.23761985990899, −11.44232551084290, −11.18258993483766, −10.80452437665962, −10.14259931414928, −9.545213914168818, −9.072806854316159, −8.890077520795699, −8.307747803049630, −7.507344928394637, −6.945962197356927, −6.785732562816596, −5.970636526622670, −5.552421948840290, −4.914460596645714, −4.082383354157518, −3.593308006590984, −2.938213011932344, −2.170264779060323, −1.483377409928277, −1.052322452276960, 0, 1.052322452276960, 1.483377409928277, 2.170264779060323, 2.938213011932344, 3.593308006590984, 4.082383354157518, 4.914460596645714, 5.552421948840290, 5.970636526622670, 6.785732562816596, 6.945962197356927, 7.507344928394637, 8.307747803049630, 8.890077520795699, 9.072806854316159, 9.545213914168818, 10.14259931414928, 10.80452437665962, 11.18258993483766, 11.44232551084290, 12.23761985990899, 12.70166004256266, 13.09926284988156, 13.84382797770955, 14.12392894809639

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