Properties

Label 2-14-1.1-c13-0-4
Degree $2$
Conductor $14$
Sign $-1$
Analytic cond. $15.0123$
Root an. cond. $3.87457$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 64·2-s + 1.62e3·3-s + 4.09e3·4-s − 3.64e4·5-s − 1.04e5·6-s − 1.17e5·7-s − 2.62e5·8-s + 1.04e6·9-s + 2.32e6·10-s + 2.60e6·11-s + 6.66e6·12-s − 1.26e7·13-s + 7.52e6·14-s − 5.91e7·15-s + 1.67e7·16-s − 1.30e8·17-s − 6.71e7·18-s − 2.49e8·19-s − 1.49e8·20-s − 1.91e8·21-s − 1.66e8·22-s + 4.89e8·23-s − 4.26e8·24-s + 1.04e8·25-s + 8.07e8·26-s − 8.85e8·27-s − 4.81e8·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.28·3-s + 1/2·4-s − 1.04·5-s − 0.910·6-s − 0.377·7-s − 0.353·8-s + 0.658·9-s + 0.736·10-s + 0.443·11-s + 0.643·12-s − 0.725·13-s + 0.267·14-s − 1.34·15-s + 1/4·16-s − 1.31·17-s − 0.465·18-s − 1.21·19-s − 0.520·20-s − 0.486·21-s − 0.313·22-s + 0.688·23-s − 0.455·24-s + 0.0854·25-s + 0.512·26-s − 0.440·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-1$
Analytic conductor: \(15.0123\)
Root analytic conductor: \(3.87457\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{6} T \)
7 \( 1 + p^{6} T \)
good3 \( 1 - 542 p T + p^{13} T^{2} \)
5 \( 1 + 1456 p^{2} T + p^{13} T^{2} \)
11 \( 1 - 2605288 T + p^{13} T^{2} \)
13 \( 1 + 12624468 T + p^{13} T^{2} \)
17 \( 1 + 130752362 T + p^{13} T^{2} \)
19 \( 1 + 249436042 T + p^{13} T^{2} \)
23 \( 1 - 489054160 T + p^{13} T^{2} \)
29 \( 1 + 112115926 T + p^{13} T^{2} \)
31 \( 1 + 9103068684 T + p^{13} T^{2} \)
37 \( 1 - 18308169938 T + p^{13} T^{2} \)
41 \( 1 - 13082373606 T + p^{13} T^{2} \)
43 \( 1 + 67123460032 T + p^{13} T^{2} \)
47 \( 1 - 105239980284 T + p^{13} T^{2} \)
53 \( 1 + 25221720042 T + p^{13} T^{2} \)
59 \( 1 + 276774602098 T + p^{13} T^{2} \)
61 \( 1 - 759388645560 T + p^{13} T^{2} \)
67 \( 1 - 1039664575708 T + p^{13} T^{2} \)
71 \( 1 - 1817086195456 T + p^{13} T^{2} \)
73 \( 1 - 400342248850 T + p^{13} T^{2} \)
79 \( 1 + 3597798513336 T + p^{13} T^{2} \)
83 \( 1 + 1309030493954 T + p^{13} T^{2} \)
89 \( 1 - 1653288354570 T + p^{13} T^{2} \)
97 \( 1 + 12736909073690 T + p^{13} T^{2} \)
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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

−15.50717519594399966232332399858, −14.67046229346699721168475761155, −12.87601627344683366244344785736, −11.19695770861436642835502524317, −9.352290978282140077635738633780, −8.333209898535310523708836747723, −7.03969134084599732022202354473, −3.89084055628901776798802155104, −2.32374369602858636946108814757, 0, 2.32374369602858636946108814757, 3.89084055628901776798802155104, 7.03969134084599732022202354473, 8.333209898535310523708836747723, 9.352290978282140077635738633780, 11.19695770861436642835502524317, 12.87601627344683366244344785736, 14.67046229346699721168475761155, 15.50717519594399966232332399858

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