Properties

Label 2-2394-57.56-c1-0-2
Degree $2$
Conductor $2394$
Sign $-0.100 - 0.994i$
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  − 2-s + 4-s − 1.15i·5-s − 7-s − 8-s + 1.15i·10-s + 2.90i·11-s + 1.38i·13-s + 14-s + 16-s + 2.90i·17-s + (3.79 + 2.14i)19-s − 1.15i·20-s − 2.90i·22-s − 7.14i·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.516i·5-s − 0.377·7-s − 0.353·8-s + 0.365i·10-s + 0.876i·11-s + 0.383i·13-s + 0.267·14-s + 0.250·16-s + 0.704i·17-s + (0.870 + 0.492i)19-s − 0.258i·20-s − 0.619i·22-s − 1.49i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-0.100 - 0.994i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (1709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ -0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7596294993\)
\(L(\frac12)\) \(\approx\) \(0.7596294993\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.79 - 2.14i)T \)
good5 \( 1 + 1.15iT - 5T^{2} \)
11 \( 1 - 2.90iT - 11T^{2} \)
13 \( 1 - 1.38iT - 13T^{2} \)
17 \( 1 - 2.90iT - 17T^{2} \)
23 \( 1 + 7.14iT - 23T^{2} \)
29 \( 1 + 4.66T + 29T^{2} \)
31 \( 1 + 4.06iT - 31T^{2} \)
37 \( 1 - 6.55iT - 37T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 + 2.04T + 43T^{2} \)
47 \( 1 - 2.76iT - 47T^{2} \)
53 \( 1 + 7.39T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.29T + 61T^{2} \)
67 \( 1 - 9.91iT - 67T^{2} \)
71 \( 1 + 6.06T + 71T^{2} \)
73 \( 1 - 9.37T + 73T^{2} \)
79 \( 1 - 12.1iT - 79T^{2} \)
83 \( 1 - 14.5iT - 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 3.92iT - 97T^{2} \)
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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

−9.186931152604920604564400468320, −8.439034754138502793175270401236, −7.79834388315267894528592264477, −6.87286643968984711023345597778, −6.31293057590650373539713482598, −5.23646956155696811884194496129, −4.41029133553639128278686465259, −3.35893331756626149378883933572, −2.20330055326520204609258480559, −1.19846467469450061405261151503, 0.34897871331763152470663298116, 1.70322534066303850513589124078, 3.17246725199094981257075067431, 3.31112862158275651186014667865, 4.99846117907972895752553536937, 5.72992986208555849365122175993, 6.63143817567202720842767614865, 7.31918116266606818118979349300, 7.911142995724033212553115306251, 8.991284510765432676353832254610

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