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Properties

Label 2-1368-1368.659-c0-0-1
Degree $2$
Conductor $1368$
Sign $-0.984 - 0.174i$
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 + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s − 0.999·6-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s − 1.96i·11-s + (0.173 + 0.984i)12-s + (0.766 − 0.642i)16-s + (−0.173 + 0.984i)18-s + (−0.766 − 0.642i)19-s + (−1.93 + 0.342i)22-s + (0.939 − 0.342i)24-s + (−0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s − 0.999·6-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s − 1.96i·11-s + (0.173 + 0.984i)12-s + (0.766 − 0.642i)16-s + (−0.173 + 0.984i)18-s + (−0.766 − 0.642i)19-s + (−1.93 + 0.342i)22-s + (0.939 − 0.342i)24-s + (−0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.984 - 0.174i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.984 - 0.174i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7743588687\)
\(L(\frac12)\) \(\approx\) \(0.7743588687\)
\(L(1)\) 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 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
good5 \( 1 + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + 1.96iT - T^{2} \)
13 \( 1 + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.766 - 0.642i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \)
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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

−9.105881515177517743380693221981, −8.688360310731783264857531403130, −8.022300214597777434422926941787, −7.03551975771155303578369862716, −5.98843016606136029527073576737, −5.22003809662985117636369291257, −3.74870502634922864286496876905, −3.04773751715605697501163834444, −1.98721099452605439156374910084, −0.67447076887785906305050913062, 2.08153078797887790209909760805, 3.66886465744554193861522075947, 4.56015679700860275707656446414, 5.00652930766022166566242725150, 6.17684408069840852422202012636, 6.91070593380754344428085874008, 7.973442559200794572362064373118, 8.452912510112282702024790837614, 9.588039386005478400587362369236, 9.875001489545442128055739577294

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