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Properties

Label 2-1368-152.75-c1-0-25
Degree $2$
Conductor $1368$
Sign $0.996 + 0.0876i$
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  + (−0.388 − 1.35i)2-s + (−1.69 + 1.05i)4-s + 1.84i·5-s − 2.17i·7-s + (2.09 + 1.89i)8-s + (2.50 − 0.716i)10-s + 4.96·11-s − 3.69·13-s + (−2.96 + 0.846i)14-s + (1.76 − 3.58i)16-s + 2.85·17-s + (−3.47 + 2.63i)19-s + (−1.94 − 3.13i)20-s + (−1.92 − 6.75i)22-s + 7.16i·23-s + ⋯
L(s)  = 1  + (−0.274 − 0.961i)2-s + (−0.849 + 0.528i)4-s + 0.824i·5-s − 0.823i·7-s + (0.740 + 0.671i)8-s + (0.793 − 0.226i)10-s + 1.49·11-s − 1.02·13-s + (−0.792 + 0.226i)14-s + (0.442 − 0.896i)16-s + 0.692·17-s + (−0.796 + 0.604i)19-s + (−0.435 − 0.700i)20-s + (−0.411 − 1.43i)22-s + 1.49i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.996 + 0.0876i$
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: \((2,\ 1368,\ (\ :1/2),\ 0.996 + 0.0876i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.274008487\)
\(L(\frac12)\) \(\approx\) \(1.274008487\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.388 + 1.35i)T \)
3 \( 1 \)
19 \( 1 + (3.47 - 2.63i)T \)
good5 \( 1 - 1.84iT - 5T^{2} \)
7 \( 1 + 2.17iT - 7T^{2} \)
11 \( 1 - 4.96T + 11T^{2} \)
13 \( 1 + 3.69T + 13T^{2} \)
17 \( 1 - 2.85T + 17T^{2} \)
23 \( 1 - 7.16iT - 23T^{2} \)
29 \( 1 + 9.32T + 29T^{2} \)
31 \( 1 - 4.76T + 31T^{2} \)
37 \( 1 - 2.36T + 37T^{2} \)
41 \( 1 + 0.613iT - 41T^{2} \)
43 \( 1 + 1.93T + 43T^{2} \)
47 \( 1 - 7.16iT - 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 - 4.64iT - 59T^{2} \)
61 \( 1 - 1.26iT - 61T^{2} \)
67 \( 1 + 3.30iT - 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 - 15.7T + 73T^{2} \)
79 \( 1 - 5.39T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 3.23iT - 89T^{2} \)
97 \( 1 - 4.47iT - 97T^{2} \)
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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

−9.655700733355420789923785653289, −9.157092234712006308406291133830, −7.88805754783443874025114902557, −7.34249932150730837132062590995, −6.44231026086801926548610803205, −5.20938254428884322008775589147, −3.96991412025332165591225574676, −3.58830930694486337608825309202, −2.31298261629842104260364430929, −1.14053186893637829447070109140, 0.70126266007657558944767615803, 2.16837613589948026494975536170, 3.89147874363156102112959429590, 4.75912918273621943055566985391, 5.45725976569004990842242123806, 6.41891510449885304002509785711, 7.04119158774489592811242702711, 8.173145431715376409470367122454, 8.764023678438754550242275281120, 9.321135033161449739179721037352

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