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Properties

Label 2-2394-57.8-c1-0-38
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: $\chi_{2394} (1205, \cdot )$
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

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

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