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Properties

Label 2-1368-152.75-c1-0-20
Degree $2$
Conductor $1368$
Sign $-0.793 - 0.609i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.793 - 0.609i$
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: \((2,\ 1368,\ (\ :1/2),\ -0.793 - 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.064445060\)
\(L(\frac12)\) \(\approx\) \(1.064445060\)
\(L(\frac{3}{2})\) 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

−9.923285893691201059289815368665, −9.226119456329777355570576481141, −7.914497303958000258361895729406, −7.52867434776748346072519811365, −6.86974157479190691639156994136, −5.87941015528184231517589359742, −5.30175467921112370610646467805, −3.80095707374320968262692030787, −2.84822002522585772386851591415, −1.35650943784677351719119192969, 0.60849748528795134236792347811, 1.62710545048544053237853998831, 2.93868562369941949766613504688, 3.99674933768556655727892359853, 4.96943179187666323362880152635, 5.70100336668099121159769634702, 7.18459425463250967662098016562, 7.960532223792963311176428921344, 8.608605476210440304388920052284, 9.298037321622526582848822525681

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