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Properties

Label 2-1368-1368.139-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.775 - 0.631i$
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 + (−0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)6-s + (−0.500 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (−0.173 + 0.300i)11-s + (−0.939 + 0.342i)12-s + (0.173 + 0.984i)16-s + (1.53 − 1.28i)17-s + (0.173 + 0.984i)18-s + (0.766 + 0.642i)19-s + (0.266 − 0.223i)22-s + 24-s + (0.766 + 0.642i)25-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (0.766 − 0.642i)6-s + (−0.500 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (−0.173 + 0.300i)11-s + (−0.939 + 0.342i)12-s + (0.173 + 0.984i)16-s + (1.53 − 1.28i)17-s + (0.173 + 0.984i)18-s + (0.766 + 0.642i)19-s + (0.266 − 0.223i)22-s + 24-s + (0.766 + 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.775 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6110586148\)
\(L(\frac12)\) \(\approx\) \(0.6110586148\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 - 0.347T + T^{2} \)
89 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
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

−9.846289382140776667647137129586, −9.395465705229676889377819680891, −8.426832586535847475948917197618, −7.55977079646522791131036223693, −6.77592619028333906413875261834, −5.66506045841565804073167164308, −4.89286383570193069118792447365, −3.56587424232188166235918940197, −2.90451595872579483630719479463, −1.15927913869796972108023885925, 0.932155032862859008161388907082, 2.08700648259279201111607962034, 3.33405259242492598357121891106, 5.16511606135020667113779332743, 5.70504608035836393878162747869, 6.65260075411711102404757488422, 7.24412049464233656868783907275, 8.184163315693915419082614079336, 8.541258499626627990827897873986, 9.755394092807748162530595851818

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