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Properties

Label 2-1368-1368.949-c0-0-5
Degree $2$
Conductor $1368$
Sign $-0.939 + 0.342i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
Motivic weight $0$
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 + (0.173 − 0.984i)3-s + (−0.499 + 0.866i)4-s + (−0.939 + 0.342i)6-s + (−0.173 − 0.300i)7-s + 0.999·8-s + (−0.939 − 0.342i)9-s + (0.766 + 0.642i)12-s + (0.939 − 1.62i)13-s + (−0.173 + 0.300i)14-s + (−0.5 − 0.866i)16-s + 0.347·17-s + (0.173 + 0.984i)18-s + 19-s + (−0.326 + 0.118i)21-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.173 − 0.984i)3-s + (−0.499 + 0.866i)4-s + (−0.939 + 0.342i)6-s + (−0.173 − 0.300i)7-s + 0.999·8-s + (−0.939 − 0.342i)9-s + (0.766 + 0.642i)12-s + (0.939 − 1.62i)13-s + (−0.173 + 0.300i)14-s + (−0.5 − 0.866i)16-s + 0.347·17-s + (0.173 + 0.984i)18-s + 19-s + (−0.326 + 0.118i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.939 + 0.342i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7729063264\)
\(L(\frac12)\) \(\approx\) \(0.7729063264\)
\(L(1)\) 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 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 - T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 0.347T + T^{2} \)
23 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + 1.87T + T^{2} \)
59 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.53T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.576580305549766269982814188532, −8.489996854222697049892883071673, −7.86842904626903123138719678419, −7.43751038549653951963111828048, −6.14170557132607873367191871875, −5.36073035306325886226331588608, −3.73882753806326187913700535227, −3.17644577749683020710174888204, −1.95639541509846893245802919484, −0.78396988303895162909045645608, 1.79411827586277084634451431385, 3.45369302715605756437669359072, 4.32510638768333390484680553954, 5.25805984562855433134776962927, 6.01185618984555511201985806732, 6.87467628598714825867298094742, 7.83498028752035202497058875371, 8.855116712700232702675094277849, 9.082072484082376700705033066696, 9.885372148651876720799800673381

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