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Properties

Label 2-1368-152.75-c1-0-81
Degree $2$
Conductor $1368$
Sign $0.625 + 0.780i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 + 0.336i)2-s + (1.77 + 0.924i)4-s − 3.04i·5-s − 2.23i·7-s + (2.12 + 1.86i)8-s + (1.02 − 4.17i)10-s + 2.29·11-s + 3.09·13-s + (0.752 − 3.07i)14-s + (2.29 + 3.27i)16-s − 5.58·17-s + (−4.29 − 0.758i)19-s + (2.81 − 5.39i)20-s + (3.14 + 0.771i)22-s − 5.51i·23-s + ⋯
L(s)  = 1  + (0.971 + 0.237i)2-s + (0.886 + 0.462i)4-s − 1.36i·5-s − 0.845i·7-s + (0.751 + 0.659i)8-s + (0.323 − 1.32i)10-s + 0.691·11-s + 0.859·13-s + (0.201 − 0.820i)14-s + (0.573 + 0.819i)16-s − 1.35·17-s + (−0.984 − 0.174i)19-s + (0.628 − 1.20i)20-s + (0.671 + 0.164i)22-s − 1.14i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.625 + 0.780i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.200214601\)
\(L(\frac12)\) \(\approx\) \(3.200214601\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.336i)T \)
3 \( 1 \)
19 \( 1 + (4.29 + 0.758i)T \)
good5 \( 1 + 3.04iT - 5T^{2} \)
7 \( 1 + 2.23iT - 7T^{2} \)
11 \( 1 - 2.29T + 11T^{2} \)
13 \( 1 - 3.09T + 13T^{2} \)
17 \( 1 + 5.58T + 17T^{2} \)
23 \( 1 + 5.51iT - 23T^{2} \)
29 \( 1 - 1.85T + 29T^{2} \)
31 \( 1 - 4.25T + 31T^{2} \)
37 \( 1 + 1.24T + 37T^{2} \)
41 \( 1 + 8.33iT - 41T^{2} \)
43 \( 1 - 4.87T + 43T^{2} \)
47 \( 1 - 9.36iT - 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + 2.49iT - 59T^{2} \)
61 \( 1 + 2.26iT - 61T^{2} \)
67 \( 1 - 4.49iT - 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 - 1.51T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 - 5.29iT - 89T^{2} \)
97 \( 1 - 1.17iT - 97T^{2} \)
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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

−9.217211472018088136548812223157, −8.591299711872111067422649714921, −7.88598687375058977362078772363, −6.64012034332506175727729225891, −6.31551822091619936382336830855, −5.01603211109840637267955727399, −4.36016817848355593684473893836, −3.84954677915041087244712285080, −2.27640367519227128612305549158, −0.992951930922282758194273496236, 1.80484369011640605973753871355, 2.71183876263683239398026935271, 3.58084917853337387672094042373, 4.46083642926551163493894937083, 5.67663345955232880603683103412, 6.50944787024663757134735269457, 6.70874100581963838099921730131, 7.966849254152057777686276449376, 8.979239936114542836026647783256, 9.907953149108977362402598211947

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