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Properties

Label 2-1368-19.7-c1-0-24
Degree $2$
Conductor $1368$
Sign $-0.658 - 0.752i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)5-s − 3·11-s + (−1 + 1.73i)13-s + (1 + 1.73i)17-s + (0.5 − 4.33i)19-s + (3 − 5.19i)23-s + (−5.49 + 9.52i)25-s + (−2 + 3.46i)29-s − 10·31-s + 2·37-s + (4.5 + 7.79i)41-s + (2 + 3.46i)43-s + (−6 + 10.3i)47-s − 7·49-s + (−1 + 1.73i)53-s + ⋯
L(s)  = 1  + (−0.894 − 1.54i)5-s − 0.904·11-s + (−0.277 + 0.480i)13-s + (0.242 + 0.420i)17-s + (0.114 − 0.993i)19-s + (0.625 − 1.08i)23-s + (−1.09 + 1.90i)25-s + (−0.371 + 0.643i)29-s − 1.79·31-s + 0.328·37-s + (0.702 + 1.21i)41-s + (0.304 + 0.528i)43-s + (−0.875 + 1.51i)47-s − 49-s + (−0.137 + 0.237i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.658 - 0.752i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 4.33i)T \)
good5 \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.5 - 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5T + 83T^{2} \)
89 \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.107795273162901345864065081160, −8.200689517438082779594401408425, −7.72290427755242095321194105477, −6.71913611105134990652463654128, −5.43214842127881589586722106763, −4.81663906891376778980749438302, −4.11240635181886462407126773576, −2.85589525382821161172138339512, −1.34720092598821044414781378375, 0, 2.19113075987656725040316578571, 3.24943920395507697830303122771, 3.78285456924514605925214456129, 5.20209298951459580357708473155, 5.98152341278566681831237526760, 7.22401073194221916980695950331, 7.45095371135005017210275298730, 8.216124615608848005661779157040, 9.457994125107719805097877520386

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