Properties

Label 2-2394-1.1-c1-0-11
Degree $2$
Conductor $2394$
Sign $1$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 3.23·5-s − 7-s − 8-s − 3.23·10-s + 0.763·11-s + 1.23·13-s + 14-s + 16-s − 7.70·17-s + 19-s + 3.23·20-s − 0.763·22-s + 5.23·23-s + 5.47·25-s − 1.23·26-s − 28-s + 8.47·29-s − 2·31-s − 32-s + 7.70·34-s − 3.23·35-s − 10.4·37-s − 38-s − 3.23·40-s + 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.44·5-s − 0.377·7-s − 0.353·8-s − 1.02·10-s + 0.230·11-s + 0.342·13-s + 0.267·14-s + 0.250·16-s − 1.86·17-s + 0.229·19-s + 0.723·20-s − 0.162·22-s + 1.09·23-s + 1.09·25-s − 0.242·26-s − 0.188·28-s + 1.57·29-s − 0.359·31-s − 0.176·32-s + 1.32·34-s − 0.546·35-s − 1.72·37-s − 0.162·38-s − 0.511·40-s + 0.937·41-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: no
Arithmetic: yes
Character: $\chi_{2394} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.665953368\)
\(L(\frac12)\) \(\approx\) \(1.665953368\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.23T + 5T^{2} \)
11 \( 1 - 0.763T + 11T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 + 7.70T + 17T^{2} \)
23 \( 1 - 5.23T + 23T^{2} \)
29 \( 1 - 8.47T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 10.4T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 - 8.94T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 0.472T + 61T^{2} \)
67 \( 1 - 0.291T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 0.763T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 4.76T + 97T^{2} \)
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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

−8.858885999448037013391045226848, −8.715010433521129373296102434571, −7.22325951581538230390779888314, −6.72237546101772004308449263692, −6.01899377163174968850444715338, −5.24405013490874731874129327005, −4.12144236108976087326420574675, −2.77473240337934365991383204539, −2.11753775817860732627324061317, −0.941750525556653715752685405721, 0.941750525556653715752685405721, 2.11753775817860732627324061317, 2.77473240337934365991383204539, 4.12144236108976087326420574675, 5.24405013490874731874129327005, 6.01899377163174968850444715338, 6.72237546101772004308449263692, 7.22325951581538230390779888314, 8.715010433521129373296102434571, 8.858885999448037013391045226848

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