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Properties

Label 2-1368-152.75-c1-0-37
Degree $2$
Conductor $1368$
Sign $-0.974 - 0.222i$
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  + (0.847 + 1.13i)2-s + (−0.562 + 1.91i)4-s + 2.55i·5-s + 3.08i·7-s + (−2.64 + 0.990i)8-s + (−2.88 + 2.16i)10-s + 5.76·11-s + 1.95·13-s + (−3.49 + 2.61i)14-s + (−3.36 − 2.15i)16-s + 1.39·17-s + (3.64 + 2.39i)19-s + (−4.89 − 1.43i)20-s + (4.88 + 6.52i)22-s + 1.59i·23-s + ⋯
L(s)  = 1  + (0.599 + 0.800i)2-s + (−0.281 + 0.959i)4-s + 1.14i·5-s + 1.16i·7-s + (−0.936 + 0.350i)8-s + (−0.913 + 0.684i)10-s + 1.73·11-s + 0.541·13-s + (−0.934 + 0.700i)14-s + (−0.841 − 0.539i)16-s + 0.339·17-s + (0.835 + 0.550i)19-s + (−1.09 − 0.321i)20-s + (1.04 + 1.39i)22-s + 0.331i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.974 - 0.222i$
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: \((2,\ 1368,\ (\ :1/2),\ -0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.389947284\)
\(L(\frac12)\) \(\approx\) \(2.389947284\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.847 - 1.13i)T \)
3 \( 1 \)
19 \( 1 + (-3.64 - 2.39i)T \)
good5 \( 1 - 2.55iT - 5T^{2} \)
7 \( 1 - 3.08iT - 7T^{2} \)
11 \( 1 - 5.76T + 11T^{2} \)
13 \( 1 - 1.95T + 13T^{2} \)
17 \( 1 - 1.39T + 17T^{2} \)
23 \( 1 - 1.59iT - 23T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 + 4.85T + 31T^{2} \)
37 \( 1 - 11.2T + 37T^{2} \)
41 \( 1 + 6.86iT - 41T^{2} \)
43 \( 1 + 8.37T + 43T^{2} \)
47 \( 1 + 8.93iT - 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + 12.8iT - 59T^{2} \)
61 \( 1 - 6.56iT - 61T^{2} \)
67 \( 1 + 16.2iT - 67T^{2} \)
71 \( 1 - 4.47T + 71T^{2} \)
73 \( 1 - 3.48T + 73T^{2} \)
79 \( 1 - 8.99T + 79T^{2} \)
83 \( 1 + 8.73T + 83T^{2} \)
89 \( 1 - 9.05iT - 89T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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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

−9.555130506759478523179343215946, −9.201447358067531144794590990567, −8.189472222650597487048018923155, −7.34241241326348313277016102236, −6.52809237669466632324120028965, −5.99663863365669085674456241980, −5.19212913357927469737853174495, −3.73636257967418486185008887042, −3.35782988140545146670427721706, −1.99380897954599305612337315642, 0.915887285046867334553858227749, 1.48318117502105300929501293847, 3.27422950274558600727972620411, 4.11357877161035585414134931849, 4.63158139476242209768104907283, 5.72043290356012542092426446926, 6.56798263747407692827419796181, 7.56577030718709376854352297284, 8.738989267530356077646740025409, 9.402381353252384978923809381591

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