Properties

Label 2-14-1.1-c13-0-5
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 − 1.02e3·3-s + 4.09e3·4-s + 4.32e3·5-s − 6.56e4·6-s + 1.17e5·7-s + 2.62e5·8-s − 5.41e5·9-s + 2.76e5·10-s − 8.78e6·11-s − 4.20e6·12-s − 2.04e7·13-s + 7.52e6·14-s − 4.43e6·15-s + 1.67e7·16-s + 1.71e6·17-s − 3.46e7·18-s − 1.09e8·19-s + 1.76e7·20-s − 1.20e8·21-s − 5.62e8·22-s − 6.46e8·23-s − 2.68e8·24-s − 1.20e9·25-s − 1.30e9·26-s + 2.19e9·27-s + 4.81e8·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.812·3-s + 1/2·4-s + 0.123·5-s − 0.574·6-s + 0.377·7-s + 0.353·8-s − 0.339·9-s + 0.0874·10-s − 1.49·11-s − 0.406·12-s − 1.17·13-s + 0.267·14-s − 0.100·15-s + 1/4·16-s + 0.0172·17-s − 0.240·18-s − 0.534·19-s + 0.0618·20-s − 0.307·21-s − 1.05·22-s − 0.910·23-s − 0.287·24-s − 0.984·25-s − 0.829·26-s + 1.08·27-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 \( 1 + 38 p^{3} T + p^{13} T^{2} \)
5 \( 1 - 864 p T + p^{13} T^{2} \)
11 \( 1 + 8787312 T + p^{13} T^{2} \)
13 \( 1 + 20420932 T + p^{13} T^{2} \)
17 \( 1 - 1719462 T + p^{13} T^{2} \)
19 \( 1 + 109702942 T + p^{13} T^{2} \)
23 \( 1 + 646760160 T + p^{13} T^{2} \)
29 \( 1 - 728867274 T + p^{13} T^{2} \)
31 \( 1 - 1028049116 T + p^{13} T^{2} \)
37 \( 1 - 14229390962 T + p^{13} T^{2} \)
41 \( 1 - 44544458406 T + p^{13} T^{2} \)
43 \( 1 + 54689828968 T + p^{13} T^{2} \)
47 \( 1 - 47868325716 T + p^{13} T^{2} \)
53 \( 1 + 169986882858 T + p^{13} T^{2} \)
59 \( 1 + 300765540198 T + p^{13} T^{2} \)
61 \( 1 - 369996272360 T + p^{13} T^{2} \)
67 \( 1 + 787010801908 T + p^{13} T^{2} \)
71 \( 1 - 559441472256 T + p^{13} T^{2} \)
73 \( 1 - 121137579650 T + p^{13} T^{2} \)
79 \( 1 - 290426785064 T + p^{13} T^{2} \)
83 \( 1 + 3965105603046 T + p^{13} T^{2} \)
89 \( 1 + 6025919250630 T + p^{13} T^{2} \)
97 \( 1 - 11302818199190 T + p^{13} T^{2} \)
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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

−15.63705033608913310438039885834, −14.24971356415454753954880496937, −12.75807747849829358787186789012, −11.54949828374638720530535194494, −10.26398123799017957117895697677, −7.79879227796490286493012193853, −5.92500521625767098203232889554, −4.77489662943688934543950811295, −2.45459606414312679081256500314, 0, 2.45459606414312679081256500314, 4.77489662943688934543950811295, 5.92500521625767098203232889554, 7.79879227796490286493012193853, 10.26398123799017957117895697677, 11.54949828374638720530535194494, 12.75807747849829358787186789012, 14.24971356415454753954880496937, 15.63705033608913310438039885834

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