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Properties

Label 2-1274-13.3-c1-0-43
Degree $2$
Conductor $1274$
Sign $-0.664 - 0.746i$
Analytic cond. $10.1729$
Root an. cond. $3.18950$
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  + (−0.5 − 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 + 0.866i)4-s − 0.267·5-s + (−0.866 + 1.5i)6-s + 0.999·8-s + (0.133 + 0.232i)10-s + (2.73 + 4.73i)11-s + 1.73·12-s + (−1.59 − 3.23i)13-s + (0.232 + 0.401i)15-s + (−0.5 − 0.866i)16-s + (−1.73 + 3i)17-s + (1.73 − 3i)19-s + (0.133 − 0.232i)20-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 − 0.866i)3-s + (−0.249 + 0.433i)4-s − 0.119·5-s + (−0.353 + 0.612i)6-s + 0.353·8-s + (0.0423 + 0.0733i)10-s + (0.823 + 1.42i)11-s + 0.499·12-s + (−0.443 − 0.896i)13-s + (0.0599 + 0.103i)15-s + (−0.125 − 0.216i)16-s + (−0.420 + 0.727i)17-s + (0.397 − 0.688i)19-s + (0.0299 − 0.0518i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1274\)    =    \(2 \cdot 7^{2} \cdot 13\)
Sign: $-0.664 - 0.746i$
Analytic conductor: \(10.1729\)
Root analytic conductor: \(3.18950\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1274} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1274,\ (\ :1/2),\ -0.664 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3371900908\)
\(L(\frac12)\) \(\approx\) \(0.3371900908\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
13 \( 1 + (1.59 + 3.23i)T \)
good3 \( 1 + (0.866 + 1.5i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.267T + 5T^{2} \)
11 \( 1 + (-2.73 - 4.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.73 - 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 + 3i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.23 + 7.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.46 + 7.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.535T + 31T^{2} \)
37 \( 1 + (-1.26 - 2.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.73 - 8.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.92T + 47T^{2} \)
53 \( 1 + 6.92T + 53T^{2} \)
59 \( 1 + (-1.40 + 2.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.59 + 2.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.46 + 4.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.23 + 2.13i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.535T + 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (4.26 + 7.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.53 - 2.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.548096861219190483131586095208, −8.147705328668159591129626243330, −7.65581045959884003789854815950, −6.69404580370134535919181476583, −6.07634508586911311721113917498, −4.69120279730016072173684053557, −3.93536596251554329385803279473, −2.46218258691633275197741747812, −1.55459866537868910463820765551, −0.17140445791800405818515716656, 1.67718880288822059952342137110, 3.56222498918127461911567334503, 4.21986362362549678899304618809, 5.42766321220581778461123444281, 5.80370785008169480137898773699, 6.96601347202568421865889777285, 7.67955005551028282383327546636, 8.728598131008329614038900167581, 9.434464277881466465135706788718, 9.911688089639360273696337717686

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