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Properties

Label 2-1368-1368.43-c0-0-0
Degree $2$
Conductor $1368$
Sign $0.672 + 0.740i$
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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.939 + 1.62i)11-s + (0.766 + 0.642i)12-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)18-s + (0.173 + 0.984i)19-s + (1.76 − 0.642i)22-s + (0.766 − 0.642i)24-s + (−0.939 − 0.342i)25-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)3-s + (−0.939 − 0.342i)4-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (0.939 + 1.62i)11-s + (0.766 + 0.642i)12-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + (0.766 − 0.642i)18-s + (0.173 + 0.984i)19-s + (1.76 − 0.642i)22-s + (0.766 − 0.642i)24-s + (−0.939 − 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.672 + 0.740i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.672 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8176852995\)
\(L(\frac12)\) \(\approx\) \(0.8176852995\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 - 1.53T + T^{2} \)
89 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
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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

−9.946448802815514092870103464756, −9.283343855566244237165302846319, −7.995668238487636919704019686477, −7.24291691238674108458282497035, −6.19372255851319420172912250022, −5.39535408807752026760099989126, −4.49263389173918037790624673960, −3.76369564855217006142081597652, −2.19412731308593010701698740868, −1.28804791129053155134737252399, 0.904157356024648888862290169885, 3.36508521032208310347871344881, 4.03523879772761075579881312646, 5.14612250599631097674912887442, 5.83190570390503190802118161518, 6.40426662565271326661424893708, 7.26320017679782863794898057589, 8.232097304678992475345961269771, 9.063312039134347468797932158486, 9.707412206923704880924649843306

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