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Properties

Label 2-14-7.6-c12-0-4
Degree $2$
Conductor $14$
Sign $0.932 - 0.361i$
Analytic cond. $12.7959$
Root an. cond. $3.57713$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 45.2·2-s + 1.26e3i·3-s + 2.04e3·4-s − 2.31e4i·5-s − 5.72e4i·6-s + (1.09e5 − 4.25e4i)7-s − 9.26e4·8-s − 1.07e6·9-s + 1.04e6i·10-s + 2.51e6·11-s + 2.59e6i·12-s − 4.20e6i·13-s + (−4.96e6 + 1.92e6i)14-s + 2.93e7·15-s + 4.19e6·16-s − 2.05e6i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73i·3-s + 0.500·4-s − 1.48i·5-s − 1.22i·6-s + (0.932 − 0.361i)7-s − 0.353·8-s − 2.01·9-s + 1.04i·10-s + 1.41·11-s + 0.868i·12-s − 0.870i·13-s + (−0.659 + 0.255i)14-s + 2.57·15-s + 0.250·16-s − 0.0849i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.932 - 0.361i$
Analytic conductor: \(12.7959\)
Root analytic conductor: \(3.57713\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :6),\ 0.932 - 0.361i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.43747 + 0.269093i\)
\(L(\frac12)\) \(\approx\) \(1.43747 + 0.269093i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 45.2T \)
7 \( 1 + (-1.09e5 + 4.25e4i)T \)
good3 \( 1 - 1.26e3iT - 5.31e5T^{2} \)
5 \( 1 + 2.31e4iT - 2.44e8T^{2} \)
11 \( 1 - 2.51e6T + 3.13e12T^{2} \)
13 \( 1 + 4.20e6iT - 2.32e13T^{2} \)
17 \( 1 + 2.05e6iT - 5.82e14T^{2} \)
19 \( 1 - 5.17e7iT - 2.21e15T^{2} \)
23 \( 1 - 1.47e8T + 2.19e16T^{2} \)
29 \( 1 - 6.75e8T + 3.53e17T^{2} \)
31 \( 1 + 2.35e8iT - 7.87e17T^{2} \)
37 \( 1 + 1.94e9T + 6.58e18T^{2} \)
41 \( 1 + 4.99e9iT - 2.25e19T^{2} \)
43 \( 1 - 5.52e9T + 3.99e19T^{2} \)
47 \( 1 - 4.82e9iT - 1.16e20T^{2} \)
53 \( 1 - 2.56e10T + 4.91e20T^{2} \)
59 \( 1 - 8.26e9iT - 1.77e21T^{2} \)
61 \( 1 + 3.69e10iT - 2.65e21T^{2} \)
67 \( 1 - 7.26e8T + 8.18e21T^{2} \)
71 \( 1 - 1.17e11T + 1.64e22T^{2} \)
73 \( 1 + 4.53e10iT - 2.29e22T^{2} \)
79 \( 1 + 1.42e11T + 5.90e22T^{2} \)
83 \( 1 - 5.47e11iT - 1.06e23T^{2} \)
89 \( 1 + 6.34e11iT - 2.46e23T^{2} \)
97 \( 1 + 1.64e11iT - 6.93e23T^{2} \)
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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

−16.83362410074532445661766934198, −15.68610309823129467540331351240, −14.40471487678874209902688136296, −12.00613174787476452730695288050, −10.59293759330411789257578470132, −9.280737373771191058840515761117, −8.377928080826287042716535775956, −5.28253930833034712581853070817, −4.01160572555889778942883489151, −1.00521955906476333045225683609, 1.26582365170144037925615171057, 2.54441916295794901770372679418, 6.52154377695030563136281825494, 7.17387396880174253520652361043, 8.763594600135168324081631566690, 11.15558352217933613538549662707, 11.91609673224567081262145780183, 13.93552129777115779288779541858, 14.82183730724593043644040652124, 17.28541718215016464301382845944

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