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Properties

Label 2-1368-152.75-c1-0-54
Degree $2$
Conductor $1368$
Sign $-0.197 + 0.980i$
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  + (−0.612 − 1.27i)2-s + (−1.25 + 1.56i)4-s + 4.23i·5-s − 3.84i·7-s + (2.75 + 0.637i)8-s + (5.39 − 2.59i)10-s − 2.95·11-s − 1.09·13-s + (−4.90 + 2.35i)14-s + (−0.873 − 3.90i)16-s − 2.81·17-s + (−0.124 − 4.35i)19-s + (−6.61 − 5.29i)20-s + (1.80 + 3.76i)22-s − 2.47i·23-s + ⋯
L(s)  = 1  + (−0.432 − 0.901i)2-s + (−0.625 + 0.780i)4-s + 1.89i·5-s − 1.45i·7-s + (0.974 + 0.225i)8-s + (1.70 − 0.819i)10-s − 0.889·11-s − 0.304·13-s + (−1.31 + 0.629i)14-s + (−0.218 − 0.975i)16-s − 0.682·17-s + (−0.0285 − 0.999i)19-s + (−1.47 − 1.18i)20-s + (0.385 + 0.802i)22-s − 0.516i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.197 + 0.980i$
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: \((2,\ 1368,\ (\ :1/2),\ -0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8511479137\)
\(L(\frac12)\) \(\approx\) \(0.8511479137\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.612 + 1.27i)T \)
3 \( 1 \)
19 \( 1 + (0.124 + 4.35i)T \)
good5 \( 1 - 4.23iT - 5T^{2} \)
7 \( 1 + 3.84iT - 7T^{2} \)
11 \( 1 + 2.95T + 11T^{2} \)
13 \( 1 + 1.09T + 13T^{2} \)
17 \( 1 + 2.81T + 17T^{2} \)
23 \( 1 + 2.47iT - 23T^{2} \)
29 \( 1 - 7.63T + 29T^{2} \)
31 \( 1 - 7.90T + 31T^{2} \)
37 \( 1 - 0.594T + 37T^{2} \)
41 \( 1 + 5.98iT - 41T^{2} \)
43 \( 1 - 8.62T + 43T^{2} \)
47 \( 1 + 8.38iT - 47T^{2} \)
53 \( 1 - 1.76T + 53T^{2} \)
59 \( 1 - 7.19iT - 59T^{2} \)
61 \( 1 + 1.92iT - 61T^{2} \)
67 \( 1 + 11.9iT - 67T^{2} \)
71 \( 1 + 2.10T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 4.29T + 79T^{2} \)
83 \( 1 + 2.76T + 83T^{2} \)
89 \( 1 + 14.7iT - 89T^{2} \)
97 \( 1 - 4.36iT - 97T^{2} \)
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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

−9.830679192540054498042053529463, −8.580903782001383910835871612692, −7.61381032487287843003672691429, −7.13378672499929785433232201770, −6.42159102965976079537843065820, −4.76832649812303449591498521977, −3.94850359571896747703892942695, −2.92174591041257104904598394152, −2.36322925518686794008272280639, −0.46534548645995111565431803359, 1.12097670867041491290829787507, 2.41358382351514626845159267750, 4.35564909199977001883762822708, 4.99330173590191723406635920965, 5.64769890188849817209467509919, 6.30993063742169915782225976911, 7.80245345882047652126665972158, 8.267383764974872175271102202449, 8.825804639996912504392928664062, 9.513767418753256126392319552250

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