Properties

Label 2-2394-1.1-c1-0-43
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 − 2·11-s − 4·13-s − 14-s + 16-s − 19-s − 2·22-s − 2·23-s − 5·25-s − 4·26-s − 28-s − 6·29-s − 4·31-s + 32-s − 6·37-s − 38-s + 6·41-s − 2·44-s − 2·46-s + 4·47-s + 49-s − 5·50-s − 4·52-s + 10·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.603·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.229·19-s − 0.426·22-s − 0.417·23-s − 25-s − 0.784·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 0.986·37-s − 0.162·38-s + 0.937·41-s − 0.301·44-s − 0.294·46-s + 0.583·47-s + 1/7·49-s − 0.707·50-s − 0.554·52-s + 1.37·53-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: Trivial
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.552883285457483019151440390549, −7.39420774947384291991380176056, −7.27933586176350623461291056392, −5.96324113567297444862623324380, −5.53344493612271351602611461765, −4.53902655446390144799157235904, −3.75671759011358733432146880831, −2.74221403772722320427902199907, −1.90727701578356598386174083624, 0, 1.90727701578356598386174083624, 2.74221403772722320427902199907, 3.75671759011358733432146880831, 4.53902655446390144799157235904, 5.53344493612271351602611461765, 5.96324113567297444862623324380, 7.27933586176350623461291056392, 7.39420774947384291991380176056, 8.552883285457483019151440390549

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