Properties

Label 2-83790-1.1-c1-0-151
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 $2$

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 − 4·23-s + 25-s + 2·26-s − 6·29-s − 4·31-s − 32-s − 2·34-s − 6·37-s − 38-s + 40-s + 10·41-s − 4·43-s − 4·44-s + 4·46-s − 12·47-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 − 0.834·23-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 − 0.986·37-s − 0.162·38-s + 0.158·40-s + 1.56·41-s − 0.609·43-s − 0.603·44-s + 0.589·46-s − 1.75·47-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: $\chi_{83790} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.51222066721497, −14.07287096837761, −13.35197052072748, −12.77093985223054, −12.52911584631833, −11.89279537977242, −11.32296112585882, −11.00138578606255, −10.34808934609644, −9.986577427941002, −9.457079747679905, −8.962029393788038, −8.219897852642171, −7.919700679432117, −7.404740721071735, −7.105975413273134, −6.176045122629032, −5.798532453280269, −5.057260922912950, −4.693292875567288, −3.709656773436253, −3.307067176560663, −2.571707892822194, −1.965258118860357, −1.259802285035381, 0, 0, 1.259802285035381, 1.965258118860357, 2.571707892822194, 3.307067176560663, 3.709656773436253, 4.693292875567288, 5.057260922912950, 5.798532453280269, 6.176045122629032, 7.105975413273134, 7.404740721071735, 7.919700679432117, 8.219897852642171, 8.962029393788038, 9.457079747679905, 9.986577427941002, 10.34808934609644, 11.00138578606255, 11.32296112585882, 11.89279537977242, 12.52911584631833, 12.77093985223054, 13.35197052072748, 14.07287096837761, 14.51222066721497

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