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Properties

Label 2-1368-1368.1075-c0-0-3
Degree $2$
Conductor $1368$
Sign $0.377 + 0.925i$
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  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)6-s + 8-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + (−0.5 − 0.866i)12-s + 16-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + 19-s + (−1 − 1.73i)22-s + (−0.5 − 0.866i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)6-s + 8-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + (−0.5 − 0.866i)12-s + 16-s + (0.5 − 0.866i)17-s + (−0.499 + 0.866i)18-s + 19-s + (−1 − 1.73i)22-s + (−0.5 − 0.866i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.377 + 0.925i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.377 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.675740766\)
\(L(\frac12)\) \(\approx\) \(1.675740766\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.864479568909705467475486134678, −8.504353236912691631248548878968, −7.70703983486328758816680507904, −7.16891998039505855706247864239, −6.01995005770348045884318893545, −5.60612457448574239441537681935, −4.85566921462868626831538754276, −3.32379899449556343526853567375, −2.70968048426587108429779118541, −1.19095235372130077440993126435, 1.96570688891125341170354393523, 3.17504849756498667230766547579, 4.13335733852875406146776655894, 4.90040998567246129818852550110, 5.50356808072668493089014499172, 6.42355668609849813766621903921, 7.35058419738252049834606447643, 8.119422975356664011586898569850, 9.488287464231363701618401999262, 10.23964752453507542887075538181

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