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Properties

Label 2-2394-57.50-c1-0-3
Degree $2$
Conductor $2394$
Sign $-0.736 + 0.676i$
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 + (−2.72 + 1.57i)5-s + 7-s − 0.999·8-s + (−2.72 − 1.57i)10-s + 1.09i·11-s + (3 + 1.73i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−2.05 + 1.18i)17-s + (−0.5 + 4.33i)19-s − 3.14i·20-s + (−0.949 + 0.548i)22-s + (0.550 + 0.317i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−1.21 + 0.703i)5-s + 0.377·7-s − 0.353·8-s + (−0.861 − 0.497i)10-s + 0.330i·11-s + (0.832 + 0.480i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.497 + 0.287i)17-s + (−0.114 + 0.993i)19-s − 0.703i·20-s + (−0.202 + 0.116i)22-s + (0.114 + 0.0662i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-0.736 + 0.676i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ -0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7355194043\)
\(L(\frac12)\) \(\approx\) \(0.7355194043\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.72 - 1.57i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.09iT - 11T^{2} \)
13 \( 1 + (-3 - 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.05 - 1.18i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.550 - 0.317i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.67 - 6.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 2.51iT - 37T^{2} \)
41 \( 1 + (3.94 + 6.84i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.22 - 5.58i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.12 + 3.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.22 - 2.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.550 - 0.953i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.55 - 2.68i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.02 + 4.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.89 + 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6 - 3.46i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.65iT - 83T^{2} \)
89 \( 1 + (0.949 - 1.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.34 - 4.24i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.136632540966265284243489163265, −8.484893052527445522230186833153, −7.72921206285010031980937968840, −7.21202686800626954028352377989, −6.44275900234660704288863668565, −5.62481830268956273229446293381, −4.51690959450779595826340786113, −3.89305218970925027697875112736, −3.20245457035456001731236666262, −1.75368025902883813475066299922, 0.23348430358978960315335738370, 1.31723942714824743017401401724, 2.72749133310527732589422660523, 3.65467749295113135804823049637, 4.40935534947337102013351788652, 5.02203731132555292827490234423, 5.98998058254459746421769391024, 6.99410556258477418984250159264, 7.955508337207359862616812074491, 8.507055974146734267468072185545

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