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Properties

Label 2-1368-152.75-c1-0-78
Degree $2$
Conductor $1368$
Sign $0.622 + 0.782i$
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  + (1.41 − 0.0800i)2-s + (1.98 − 0.226i)4-s + 1.12i·5-s − 4.22i·7-s + (2.78 − 0.478i)8-s + (0.0900 + 1.58i)10-s − 2.32·11-s − 1.28·13-s + (−0.338 − 5.96i)14-s + (3.89 − 0.898i)16-s + 4.98·17-s + (−2.09 − 3.82i)19-s + (0.254 + 2.23i)20-s + (−3.28 + 0.186i)22-s − 7.45i·23-s + ⋯
L(s)  = 1  + (0.998 − 0.0566i)2-s + (0.993 − 0.113i)4-s + 0.503i·5-s − 1.59i·7-s + (0.985 − 0.169i)8-s + (0.0284 + 0.502i)10-s − 0.702·11-s − 0.357·13-s + (−0.0903 − 1.59i)14-s + (0.974 − 0.224i)16-s + 1.20·17-s + (−0.480 − 0.876i)19-s + (0.0568 + 0.499i)20-s + (−0.700 + 0.0397i)22-s − 1.55i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.622 + 0.782i$
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: \((2,\ 1368,\ (\ :1/2),\ 0.622 + 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.176022709\)
\(L(\frac12)\) \(\approx\) \(3.176022709\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0800i)T \)
3 \( 1 \)
19 \( 1 + (2.09 + 3.82i)T \)
good5 \( 1 - 1.12iT - 5T^{2} \)
7 \( 1 + 4.22iT - 7T^{2} \)
11 \( 1 + 2.32T + 11T^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
17 \( 1 - 4.98T + 17T^{2} \)
23 \( 1 + 7.45iT - 23T^{2} \)
29 \( 1 - 3.42T + 29T^{2} \)
31 \( 1 - 8.74T + 31T^{2} \)
37 \( 1 + 5.87T + 37T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 - 5.81iT - 47T^{2} \)
53 \( 1 - 4.80T + 53T^{2} \)
59 \( 1 - 3.68iT - 59T^{2} \)
61 \( 1 + 8.19iT - 61T^{2} \)
67 \( 1 - 4.42iT - 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 8.02T + 73T^{2} \)
79 \( 1 - 5.34T + 79T^{2} \)
83 \( 1 + 4.86T + 83T^{2} \)
89 \( 1 + 3.70iT - 89T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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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

−10.14547656026664258884752874550, −8.355647000410519979810918760360, −7.62571808568612376734504837876, −6.81344654369067979689593320421, −6.37132574047880447517304253592, −4.92279103283098832726485752438, −4.52419732996030425105320301747, −3.32760071261362965551332551499, −2.63360707078496390250935743254, −0.981313405021033413315467392164, 1.68120262795218897109751163473, 2.71616366561184820789827372482, 3.58682803528403590352336361715, 4.97554756413543857747782299012, 5.37789701118760167260053384464, 6.06725585476058827696269058485, 7.16164791562510557613178755432, 8.134679042162083853921519755040, 8.671873763024764204199200013944, 9.870094445905394442785665108249

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