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Properties

Label 2-1368-152.75-c1-0-62
Degree $2$
Conductor $1368$
Sign $0.929 - 0.369i$
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  + (0.906 + 1.08i)2-s + (−0.356 + 1.96i)4-s − 3.19i·5-s + 0.607i·7-s + (−2.45 + 1.39i)8-s + (3.47 − 2.89i)10-s − 0.581·11-s + 0.146·13-s + (−0.659 + 0.550i)14-s + (−3.74 − 1.40i)16-s + 6.67·17-s + (4.31 + 0.600i)19-s + (6.29 + 1.13i)20-s + (−0.527 − 0.631i)22-s − 5.86i·23-s + ⋯
L(s)  = 1  + (0.641 + 0.767i)2-s + (−0.178 + 0.984i)4-s − 1.43i·5-s + 0.229i·7-s + (−0.869 + 0.494i)8-s + (1.09 − 0.916i)10-s − 0.175·11-s + 0.0407·13-s + (−0.176 + 0.147i)14-s + (−0.936 − 0.350i)16-s + 1.61·17-s + (0.990 + 0.137i)19-s + (1.40 + 0.254i)20-s + (−0.112 − 0.134i)22-s − 1.22i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.929 - 0.369i$
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: \((2,\ 1368,\ (\ :1/2),\ 0.929 - 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.339245510\)
\(L(\frac12)\) \(\approx\) \(2.339245510\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.906 - 1.08i)T \)
3 \( 1 \)
19 \( 1 + (-4.31 - 0.600i)T \)
good5 \( 1 + 3.19iT - 5T^{2} \)
7 \( 1 - 0.607iT - 7T^{2} \)
11 \( 1 + 0.581T + 11T^{2} \)
13 \( 1 - 0.146T + 13T^{2} \)
17 \( 1 - 6.67T + 17T^{2} \)
23 \( 1 + 5.86iT - 23T^{2} \)
29 \( 1 - 8.81T + 29T^{2} \)
31 \( 1 + 7.41T + 31T^{2} \)
37 \( 1 - 7.25T + 37T^{2} \)
41 \( 1 - 3.86iT - 41T^{2} \)
43 \( 1 - 5.41T + 43T^{2} \)
47 \( 1 + 4.14iT - 47T^{2} \)
53 \( 1 - 5.95T + 53T^{2} \)
59 \( 1 + 9.46iT - 59T^{2} \)
61 \( 1 + 7.47iT - 61T^{2} \)
67 \( 1 - 10.5iT - 67T^{2} \)
71 \( 1 + 4.38T + 71T^{2} \)
73 \( 1 + 7.30T + 73T^{2} \)
79 \( 1 + 1.45T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + 2.46iT - 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−9.371814621779735449697367387115, −8.625306140542996772951621866165, −8.030000447748054767831369338720, −7.27512652283827185890932158185, −6.09026386095660699947039364391, −5.39764530088651380238142072527, −4.78931016637340541067838206131, −3.85524342146700857846935600698, −2.70978973634061893122749420057, −0.960385601128885990583550860722, 1.21451779514603417672608959779, 2.69552109450985484700396032343, 3.24427230243485952977628103477, 4.14707274691127921443030306310, 5.46292037119779951838465799634, 5.97191172831731222640309610725, 7.15285431613988072524575369300, 7.59616484084873988534484297301, 9.042084664876378551428623247554, 9.961761726747679251975483227441

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