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Properties

Label 2-2394-19.7-c1-0-4
Degree $2$
Conductor $2394$
Sign $0.910 - 0.412i$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.5 − 2.59i)5-s − 7-s + 0.999·8-s + (−1.5 + 2.59i)10-s − 4·11-s + (−1.5 + 2.59i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1 − 1.73i)17-s + (−0.5 − 4.33i)19-s + 3·20-s + (2 + 3.46i)22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.670 − 1.16i)5-s − 0.377·7-s + 0.353·8-s + (−0.474 + 0.821i)10-s − 1.20·11-s + (−0.416 + 0.720i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + (−0.114 − 0.993i)19-s + 0.670·20-s + (0.426 + 0.738i)22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 0.910 - 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4205979085\)
\(L(\frac12)\) \(\approx\) \(0.4205979085\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + T \)
19 \( 1 + (0.5 + 4.33i)T \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-2 - 3.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 - 5.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.5 - 11.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.5 + 6.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17T + 83T^{2} \)
89 \( 1 + (-2 + 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.133970303705775399907781801249, −8.380985307150271879563876966087, −7.65775925697095361332967329054, −6.98274740298103474054795386629, −5.71827095948449913959230065037, −4.72005707002373851847232298019, −4.37195260222924538857240390963, −3.10850891205835588942920262761, −2.23421797727027978624717093921, −0.814600805322982599120281158000, 0.21866548320714900126141631064, 2.17906795911070252714031494425, 3.18348530336096331408348201154, 3.96268734758562281318611646834, 5.22089256028457055065771119046, 5.87346395470072600320149432101, 6.78065127179588594301743962305, 7.47686521122505276793023086036, 7.903842742194320840529669973278, 8.703384850718296031286339079443

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