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Properties

Label 2-2394-19.7-c1-0-39
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 + (0.5 + 0.866i)5-s + 7-s − 0.999·8-s + (−0.499 + 0.866i)10-s + 2·11-s + (2.5 − 4.33i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 4.33i)19-s − 0.999·20-s + (1 + 1.73i)22-s + (0.5 − 0.866i)23-s + (2 − 3.46i)25-s + 5·26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 0.377·7-s − 0.353·8-s + (−0.158 + 0.273i)10-s + 0.603·11-s + (0.693 − 1.20i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.114 − 0.993i)19-s − 0.223·20-s + (0.213 + 0.369i)22-s + (0.104 − 0.180i)23-s + (0.400 − 0.692i)25-s + 0.980·26-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

\(L(1)\) \(\approx\) \(2.447816345\)
\(L(\frac12)\) \(\approx\) \(2.447816345\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.5 + 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6 + 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.766049285255243972653725299581, −8.225152873479505353024397754354, −7.44152417481062794098920154438, −6.51139073944148489943379009259, −6.09295100444871272002744588399, −5.08082397176612269558228985475, −4.35704175546473627047984511318, −3.32077157908290356468073734633, −2.45930340983725361587408871227, −0.859702679339627593443603714311, 1.23805533134719575055397976955, 1.87308658470553931678920740379, 3.23782375523766038743011811500, 4.06467640964997724175119809354, 4.79265747391555046582614016045, 5.68158016670952224657124663370, 6.46720097529748095166023875460, 7.28619362939549886473334309132, 8.481044083804738850281600738480, 8.923366005081995561376678456108

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