Properties

Label 2-1368-456.125-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.843 - 0.536i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s − 7-s + (−0.707 + 0.707i)8-s − 1.41·11-s + (0.866 − 0.5i)13-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)16-s + (1.22 + 0.707i)17-s + i·19-s + (1.36 − 0.366i)22-s + (1.22 − 0.707i)23-s + (0.5 + 0.866i)25-s + (−0.707 + 0.707i)26-s + (−0.866 + 0.499i)28-s + (0.707 + 1.22i)29-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s − 7-s + (−0.707 + 0.707i)8-s − 1.41·11-s + (0.866 − 0.5i)13-s + (0.965 − 0.258i)14-s + (0.500 − 0.866i)16-s + (1.22 + 0.707i)17-s + i·19-s + (1.36 − 0.366i)22-s + (1.22 − 0.707i)23-s + (0.5 + 0.866i)25-s + (−0.707 + 0.707i)26-s + (−0.866 + 0.499i)28-s + (0.707 + 1.22i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.843 - 0.536i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.843 - 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6292470817\)
\(L(\frac12)\) \(\approx\) \(0.6292470817\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
19 \( 1 - iT \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.894311732351172483761111482226, −9.001120611892888644314277596630, −8.246225643592222438984009730036, −7.60713749249786512735225932531, −6.68563742132531616817595221244, −5.86464602105852591747711116144, −5.20465116044033444610751551760, −3.43896730823926328716090196817, −2.75534878326116380895287480850, −1.13076255205986667565091005338, 0.898426640689715484740847367470, 2.72317405135435340491006686293, 3.08273986455522512506891148363, 4.57625162936233071686315920350, 5.82913072583543453679128571114, 6.59945064505376926243309844701, 7.44560688856309073608482966502, 8.145548279857054338904649111121, 9.070238794625015388191791303176, 9.662441264507527111835843794369

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