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Properties

Label 2-1368-19.7-c1-0-23
Degree $2$
Conductor $1368$
Sign $-0.910 + 0.412i$
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 − 1.73i)5-s + 3·7-s − 6·11-s + (0.5 − 0.866i)13-s + (−1 − 1.73i)17-s + (−4 − 1.73i)19-s + (0.500 − 0.866i)25-s + (−1 + 1.73i)29-s − 31-s + (−3 − 5.19i)35-s − 7·37-s + (0.5 + 0.866i)43-s + 2·49-s + (−2 + 3.46i)53-s + (6 + 10.3i)55-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + 1.13·7-s − 1.80·11-s + (0.138 − 0.240i)13-s + (−0.242 − 0.420i)17-s + (−0.917 − 0.397i)19-s + (0.100 − 0.173i)25-s + (−0.185 + 0.321i)29-s − 0.179·31-s + (−0.507 − 0.878i)35-s − 1.15·37-s + (0.0762 + 0.132i)43-s + 0.285·49-s + (−0.274 + 0.475i)53-s + (0.809 + 1.40i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.910 + 0.412i$
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: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6594099459\)
\(L(\frac12)\) \(\approx\) \(0.6594099459\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 + 1.73i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 - 12.9i)T + (-33.5 - 58.0i)T^{2} \)
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 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.998168709704336046560122042400, −8.311716297648978780958570229556, −7.88285798135470726235766528744, −6.97343054171246210416975473379, −5.62754883012658754571120704636, −4.94321129437007001119596786741, −4.38393869178348063718615890204, −2.94877821156996220038676751647, −1.82054907416013604360406302184, −0.25270772619315161914226936669, 1.83160629286675538816495346508, 2.81031159691183865612239614066, 3.96110463130137320156465563036, 4.91136873809083980733563698111, 5.70990634921058613077608227380, 6.81217165422627643114563796790, 7.67231064504793062558735913754, 8.135164510251059153697413825935, 8.954002627747658894026416721498, 10.34068756620246985831017438481

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