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Properties

Label 2-14-7.6-c8-0-0
Degree $2$
Conductor $14$
Sign $-0.978 + 0.203i$
Analytic cond. $5.70330$
Root an. cond. $2.38815$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11.3·2-s + 126. i·3-s + 128.·4-s − 143. i·5-s − 1.43e3i·6-s + (−2.35e3 + 489. i)7-s − 1.44e3·8-s − 9.43e3·9-s + 1.61e3i·10-s − 1.07e4·11-s + 1.61e4i·12-s − 4.06e4i·13-s + (2.65e4 − 5.53e3i)14-s + 1.80e4·15-s + 1.63e4·16-s + 4.96e4i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.56i·3-s + 0.500·4-s − 0.228i·5-s − 1.10i·6-s + (−0.978 + 0.203i)7-s − 0.353·8-s − 1.43·9-s + 0.161i·10-s − 0.736·11-s + 0.780i·12-s − 1.42i·13-s + (0.692 − 0.144i)14-s + 0.357·15-s + 0.250·16-s + 0.594i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.203i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.978 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.978 + 0.203i$
Analytic conductor: \(5.70330\)
Root analytic conductor: \(2.38815\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :4),\ -0.978 + 0.203i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0471017 - 0.457148i\)
\(L(\frac12)\) \(\approx\) \(0.0471017 - 0.457148i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 11.3T \)
7 \( 1 + (2.35e3 - 489. i)T \)
good3 \( 1 - 126. iT - 6.56e3T^{2} \)
5 \( 1 + 143. iT - 3.90e5T^{2} \)
11 \( 1 + 1.07e4T + 2.14e8T^{2} \)
13 \( 1 + 4.06e4iT - 8.15e8T^{2} \)
17 \( 1 - 4.96e4iT - 6.97e9T^{2} \)
19 \( 1 - 2.25e5iT - 1.69e10T^{2} \)
23 \( 1 + 2.91e5T + 7.83e10T^{2} \)
29 \( 1 + 6.86e5T + 5.00e11T^{2} \)
31 \( 1 - 1.17e6iT - 8.52e11T^{2} \)
37 \( 1 + 1.16e5T + 3.51e12T^{2} \)
41 \( 1 - 4.08e6iT - 7.98e12T^{2} \)
43 \( 1 - 6.46e5T + 1.16e13T^{2} \)
47 \( 1 + 7.85e6iT - 2.38e13T^{2} \)
53 \( 1 + 7.98e6T + 6.22e13T^{2} \)
59 \( 1 - 5.68e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.04e7iT - 1.91e14T^{2} \)
67 \( 1 - 3.25e7T + 4.06e14T^{2} \)
71 \( 1 + 1.34e7T + 6.45e14T^{2} \)
73 \( 1 + 3.32e6iT - 8.06e14T^{2} \)
79 \( 1 - 1.56e7T + 1.51e15T^{2} \)
83 \( 1 - 4.45e7iT - 2.25e15T^{2} \)
89 \( 1 + 3.46e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.47e7iT - 7.83e15T^{2} \)
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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

−18.33745712964788948455092535472, −16.69896249649477395841100287234, −15.95338999729404351717370760940, −14.92622099978917466143496734537, −12.61452822494309223187078393100, −10.52757074167974710091295951007, −9.883237269673085455580043929028, −8.322945381620476980189064903073, −5.63477003052124937149897570241, −3.37856766161271038168758780583, 0.32048335661327030632103656393, 2.36827426394006627205473115982, 6.49819014705709448408703529653, 7.45054779177891356455027304432, 9.280201888079765633283944613021, 11.29028482681189161730413531218, 12.72956230224811052319299643123, 13.86284502396763844151796009414, 15.94134433959154951676590690262, 17.33361474979372298981844847825

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