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Properties

Label 2-1368-19.11-c1-0-19
Degree $2$
Conductor $1368$
Sign $0.204 + 0.978i$
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  + (1.53 − 2.65i)5-s − 2.06·7-s + 6.45·11-s + (−0.5 − 0.866i)13-s + (0.694 − 1.20i)17-s + (−3.75 − 2.20i)19-s + (1.53 + 2.65i)23-s + (−2.19 − 3.80i)25-s + (−1.75 − 3.04i)29-s + 9.45·31-s + (−3.16 + 5.47i)35-s − 2.38·37-s + (5.06 − 8.77i)41-s + (−3.03 + 5.25i)43-s + (−3 − 5.19i)47-s + ⋯
L(s)  = 1  + (0.685 − 1.18i)5-s − 0.780·7-s + 1.94·11-s + (−0.138 − 0.240i)13-s + (0.168 − 0.291i)17-s + (−0.862 − 0.506i)19-s + (0.319 + 0.553i)23-s + (−0.438 − 0.760i)25-s + (−0.326 − 0.565i)29-s + 1.69·31-s + (−0.534 + 0.925i)35-s − 0.392·37-s + (0.790 − 1.36i)41-s + (−0.462 + 0.800i)43-s + (−0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.807000954\)
\(L(\frac12)\) \(\approx\) \(1.807000954\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (3.75 + 2.20i)T \)
good5 \( 1 + (-1.53 + 2.65i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.06T + 7T^{2} \)
11 \( 1 - 6.45T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.694 + 1.20i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.53 - 2.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.75 + 3.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.45T + 31T^{2} \)
37 \( 1 + 2.38T + 37T^{2} \)
41 \( 1 + (-5.06 + 8.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.03 - 5.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.29 + 9.16i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.59 - 9.69i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.56 + 4.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.72 + 2.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.36 + 5.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.56 + 7.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.790 + 1.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 + (-5.22 - 9.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.36 - 5.83i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.324526853071403354609042561737, −8.926513923688828341322268895543, −7.940412089166861513392469589594, −6.63565009444622167399973272032, −6.28665218104516879475013906624, −5.18173306242334255396841507591, −4.35133154099774654818488091527, −3.36104643187903926829099184267, −1.92264641989444094445282057784, −0.795024086808873161380527557363, 1.48990395070094719059366673024, 2.72461051280171794247916858762, 3.58813435629921523393983516132, 4.54377455418055608054767599199, 6.14515950194102320189883226790, 6.39040408783257216095135483618, 6.96475489631035546838155460711, 8.206180811015168249646915493803, 9.207071016716004642866145968743, 9.725314856379548701678154694645

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