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Properties

Label 2-1368-1368.1339-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.944 + 0.327i$
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  + (−0.939 − 0.342i)2-s + 3-s + (0.766 + 0.642i)4-s + (−0.939 − 0.342i)6-s + (−0.500 − 0.866i)8-s + 9-s + 0.347·11-s + (0.766 + 0.642i)12-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + (−0.939 − 0.342i)18-s + (0.766 + 0.642i)19-s + (−0.326 − 0.118i)22-s + (−0.500 − 0.866i)24-s + (−0.939 + 0.342i)25-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + 3-s + (0.766 + 0.642i)4-s + (−0.939 − 0.342i)6-s + (−0.500 − 0.866i)8-s + 9-s + 0.347·11-s + (0.766 + 0.642i)12-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + (−0.939 − 0.342i)18-s + (0.766 + 0.642i)19-s + (−0.326 − 0.118i)22-s + (−0.500 − 0.866i)24-s + (−0.939 + 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.944 + 0.327i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.057326657\)
\(L(\frac12)\) \(\approx\) \(1.057326657\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 - T \)
19 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 - 0.347T + T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.173 - 0.984i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
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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

−9.463964477065985139930029797403, −9.172499390441546929120712464767, −8.194053232253698939089370993575, −7.52894145522254680623676109656, −6.93552878311931629290211801109, −5.75309293635175892666799738823, −4.28381027611245566600458924389, −3.38626113087263582932418115737, −2.48896422121217798282457803573, −1.37812690684402521232657985478, 1.43164168372506385630254904679, 2.48368565352088062021267999577, 3.58658584976803941919363871206, 4.75983082028672499142384832834, 6.01720066092493480435067574121, 6.77606976702368184418140180853, 7.70549995224751697165440002550, 8.162844698632471218921117496591, 9.094481989113774575223253923513, 9.537450023978616118920145028996

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