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Properties

Label 2-1368-152.123-c0-0-0
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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.5 + 0.866i)8-s + (0.766 + 1.32i)11-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)32-s + (0.939 − 0.342i)34-s + (0.5 + 0.866i)38-s + (1.43 − 1.20i)41-s + (0.939 + 0.342i)43-s + (−1.17 − 0.984i)44-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.5 + 0.866i)8-s + (0.766 + 1.32i)11-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + (−0.939 + 0.342i)19-s + (1.17 − 0.984i)22-s + (0.766 + 0.642i)25-s + (−0.766 − 0.642i)32-s + (0.939 − 0.342i)34-s + (0.5 + 0.866i)38-s + (1.43 − 1.20i)41-s + (0.939 + 0.342i)43-s + (−1.17 − 0.984i)44-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: $\chi_{1368} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.944 + 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9100432835\)
\(L(\frac12)\) \(\approx\) \(0.9100432835\)
\(L(1)\) 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 \( 1 \)
19 \( 1 + (0.939 - 0.342i)T \)
good5 \( 1 + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
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 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
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 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
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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

−9.771364813906417495643272131362, −9.144789977060353409155702329344, −8.351039223547349694130518410890, −7.46493039548645712787797255844, −6.51197585659926707961430408301, −5.36270795373674529650893321116, −4.32776739984497326455100462331, −3.75770325074225807551660191206, −2.39820065182450849368983229293, −1.47620533384632479922595983506, 0.936260874972603277636415509724, 2.86846714021961662671051209902, 4.06014770763908497093717848947, 4.87527718899260544314940461264, 5.97434646435689401033691171678, 6.43916642903390883214737042578, 7.39287233844047163327224317642, 8.217506813752072512782278837207, 8.972802786353350637828390561469, 9.442599146872639339605450384510

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