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Properties

Label 2-14-7.6-c4-0-0
Degree $2$
Conductor $14$
Sign $0.131 - 0.991i$
Analytic cond. $1.44717$
Root an. cond. $1.20298$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·2-s + 12.6i·3-s + 8.00·4-s + 23.1i·5-s − 35.8i·6-s + (6.45 − 48.5i)7-s − 22.6·8-s − 79.9·9-s − 65.6i·10-s + 191.·11-s + 101. i·12-s + 48.5i·13-s + (−18.2 + 137. i)14-s − 294.·15-s + 64.0·16-s + 181. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.40i·3-s + 0.500·4-s + 0.927i·5-s − 0.996i·6-s + (0.131 − 0.991i)7-s − 0.353·8-s − 0.987·9-s − 0.656i·10-s + 1.58·11-s + 0.704i·12-s + 0.287i·13-s + (−0.0931 + 0.700i)14-s − 1.30·15-s + 0.250·16-s + 0.629i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.131 - 0.991i$
Analytic conductor: \(1.44717\)
Root analytic conductor: \(1.20298\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :2),\ 0.131 - 0.991i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.658610 + 0.576865i\)
\(L(\frac12)\) \(\approx\) \(0.658610 + 0.576865i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
7 \( 1 + (-6.45 + 48.5i)T \)
good3 \( 1 - 12.6iT - 81T^{2} \)
5 \( 1 - 23.1iT - 625T^{2} \)
11 \( 1 - 191.T + 1.46e4T^{2} \)
13 \( 1 - 48.5iT - 2.85e4T^{2} \)
17 \( 1 - 181. iT - 8.35e4T^{2} \)
19 \( 1 + 599. iT - 1.30e5T^{2} \)
23 \( 1 + 469.T + 2.79e5T^{2} \)
29 \( 1 + 338.T + 7.07e5T^{2} \)
31 \( 1 + 267. iT - 9.23e5T^{2} \)
37 \( 1 + 668.T + 1.87e6T^{2} \)
41 \( 1 + 1.32e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.94e3T + 3.41e6T^{2} \)
47 \( 1 + 2.93e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.46e3T + 7.89e6T^{2} \)
59 \( 1 - 1.73e3iT - 1.21e7T^{2} \)
61 \( 1 + 246. iT - 1.38e7T^{2} \)
67 \( 1 + 1.07e3T + 2.01e7T^{2} \)
71 \( 1 + 2.27e3T + 2.54e7T^{2} \)
73 \( 1 - 7.10e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.01e3T + 3.89e7T^{2} \)
83 \( 1 - 1.44e3iT - 4.74e7T^{2} \)
89 \( 1 - 2.13e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.89e3iT - 8.85e7T^{2} \)
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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

−19.36426199120263942520414803584, −17.55656847425271026347388221962, −16.62778998269910397541628761587, −15.26013474635776869549307680992, −14.20327862472762734726708114811, −11.32932679614669470779169211967, −10.39234639683878339141579667076, −9.151420707955181608830999583110, −6.86385157534129198436318472719, −3.93226243411066662269045761385, 1.48434266230414316686210017856, 6.16485553035341295092316063267, 7.972395856919718186848184736214, 9.222831216604606739653748821469, 11.87669990326372847463667711754, 12.51472534075513744428891950515, 14.34243191638167463686003453415, 16.28438054647606574859857138888, 17.52571806582019984053133574274, 18.57838463438063750423440720025

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