Properties

Label 2-14-1.1-c9-0-3
Degree $2$
Conductor $14$
Sign $-1$
Analytic cond. $7.21050$
Root an. cond. $2.68523$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s − 6·3-s + 256·4-s + 560·5-s + 96·6-s − 2.40e3·7-s − 4.09e3·8-s − 1.96e4·9-s − 8.96e3·10-s − 5.41e4·11-s − 1.53e3·12-s − 1.13e5·13-s + 3.84e4·14-s − 3.36e3·15-s + 6.55e4·16-s + 6.26e3·17-s + 3.14e5·18-s + 2.57e5·19-s + 1.43e5·20-s + 1.44e4·21-s + 8.66e5·22-s − 2.66e5·23-s + 2.45e4·24-s − 1.63e6·25-s + 1.81e6·26-s + 2.35e5·27-s − 6.14e5·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0427·3-s + 1/2·4-s + 0.400·5-s + 0.0302·6-s − 0.377·7-s − 0.353·8-s − 0.998·9-s − 0.283·10-s − 1.11·11-s − 0.0213·12-s − 1.09·13-s + 0.267·14-s − 0.0171·15-s + 1/4·16-s + 0.0181·17-s + 0.705·18-s + 0.452·19-s + 0.200·20-s + 0.0161·21-s + 0.788·22-s − 0.198·23-s + 0.0151·24-s − 0.839·25-s + 0.777·26-s + 0.0854·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-1$
Analytic conductor: \(7.21050\)
Root analytic conductor: \(2.68523\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: $\chi_{14} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{4} T \)
7 \( 1 + p^{4} T \)
good3 \( 1 + 2 p T + p^{9} T^{2} \)
5 \( 1 - 112 p T + p^{9} T^{2} \)
11 \( 1 + 54152 T + p^{9} T^{2} \)
13 \( 1 + 113172 T + p^{9} T^{2} \)
17 \( 1 - 6262 T + p^{9} T^{2} \)
19 \( 1 - 257078 T + p^{9} T^{2} \)
23 \( 1 + 266000 T + p^{9} T^{2} \)
29 \( 1 - 1574714 T + p^{9} T^{2} \)
31 \( 1 + 4637484 T + p^{9} T^{2} \)
37 \( 1 + 11946238 T + p^{9} T^{2} \)
41 \( 1 - 21909126 T + p^{9} T^{2} \)
43 \( 1 - 27520592 T + p^{9} T^{2} \)
47 \( 1 - 52927836 T + p^{9} T^{2} \)
53 \( 1 - 16221222 T + p^{9} T^{2} \)
59 \( 1 + 140509618 T + p^{9} T^{2} \)
61 \( 1 + 202963560 T + p^{9} T^{2} \)
67 \( 1 - 153734572 T + p^{9} T^{2} \)
71 \( 1 - 3938816 p T + p^{9} T^{2} \)
73 \( 1 + 404022830 T + p^{9} T^{2} \)
79 \( 1 + 130689816 T + p^{9} T^{2} \)
83 \( 1 - 420134014 T + p^{9} T^{2} \)
89 \( 1 + 469542390 T + p^{9} T^{2} \)
97 \( 1 + 872501690 T + p^{9} T^{2} \)
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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

−16.97156398772801953513680836714, −15.62762862669854132429460146263, −14.02868914399650773999320612569, −12.30375843851048355463008830546, −10.65802007288394348708776625087, −9.306427106328249338204083036979, −7.62305032881012570225622582573, −5.65468442777949473050849268788, −2.57769461149083632364105051475, 0, 2.57769461149083632364105051475, 5.65468442777949473050849268788, 7.62305032881012570225622582573, 9.306427106328249338204083036979, 10.65802007288394348708776625087, 12.30375843851048355463008830546, 14.02868914399650773999320612569, 15.62762862669854132429460146263, 16.97156398772801953513680836714

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