Properties

Label 2-2394-1.1-c1-0-33
Degree $2$
Conductor $2394$
Sign $-1$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
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 + 7-s − 8-s − 6·11-s + 2·13-s − 14-s + 16-s + 19-s + 6·22-s + 6·23-s − 5·25-s − 2·26-s + 28-s − 6·29-s + 8·31-s − 32-s − 10·37-s − 38-s + 6·41-s − 4·43-s − 6·44-s − 6·46-s − 6·47-s + 49-s + 5·50-s + 2·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.80·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.229·19-s + 1.27·22-s + 1.25·23-s − 25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 1.64·37-s − 0.162·38-s + 0.937·41-s − 0.609·43-s − 0.904·44-s − 0.884·46-s − 0.875·47-s + 1/7·49-s + 0.707·50-s + 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-1$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2394} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
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 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.490670546431863178338078429995, −7.87462094921647883942664875297, −7.33615990354905827830525466937, −6.31820508463320297458061667389, −5.44565375834713779887264721264, −4.76918027065692334568092858822, −3.42948835695246631266124027247, −2.57033400147183809405890786244, −1.47814962499483450164849405751, 0, 1.47814962499483450164849405751, 2.57033400147183809405890786244, 3.42948835695246631266124027247, 4.76918027065692334568092858822, 5.44565375834713779887264721264, 6.31820508463320297458061667389, 7.33615990354905827830525466937, 7.87462094921647883942664875297, 8.490670546431863178338078429995

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