Properties

Label 2-14-7.4-c13-0-2
Degree $2$
Conductor $14$
Sign $-0.924 - 0.382i$
Analytic cond. $15.0123$
Root an. cond. $3.87457$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (32 + 55.4i)2-s + (−244. + 423. i)3-s + (−2.04e3 + 3.54e3i)4-s + (3.40e4 + 5.90e4i)5-s − 3.13e4·6-s + (1.00e5 − 2.94e5i)7-s − 2.62e5·8-s + (6.77e5 + 1.17e6i)9-s + (−2.18e6 + 3.77e6i)10-s + (−1.01e6 + 1.75e6i)11-s + (−1.00e6 − 1.73e6i)12-s − 1.22e7·13-s + (1.95e7 − 3.85e6i)14-s − 3.33e7·15-s + (−8.38e6 − 1.45e7i)16-s + (−4.76e7 + 8.25e7i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.193 + 0.335i)3-s + (−0.249 + 0.433i)4-s + (0.975 + 1.69i)5-s − 0.274·6-s + (0.322 − 0.946i)7-s − 0.353·8-s + (0.424 + 0.735i)9-s + (−0.690 + 1.19i)10-s + (−0.172 + 0.298i)11-s + (−0.0968 − 0.167i)12-s − 0.701·13-s + (0.693 − 0.136i)14-s − 0.756·15-s + (−0.125 − 0.216i)16-s + (−0.478 + 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.382i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.924 - 0.382i$
Analytic conductor: \(15.0123\)
Root analytic conductor: \(3.87457\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :13/2),\ -0.924 - 0.382i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.415154 + 2.08964i\)
\(L(\frac12)\) \(\approx\) \(0.415154 + 2.08964i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-32 - 55.4i)T \)
7 \( 1 + (-1.00e5 + 2.94e5i)T \)
good3 \( 1 + (244. - 423. i)T + (-7.97e5 - 1.38e6i)T^{2} \)
5 \( 1 + (-3.40e4 - 5.90e4i)T + (-6.10e8 + 1.05e9i)T^{2} \)
11 \( 1 + (1.01e6 - 1.75e6i)T + (-1.72e13 - 2.98e13i)T^{2} \)
13 \( 1 + 1.22e7T + 3.02e14T^{2} \)
17 \( 1 + (4.76e7 - 8.25e7i)T + (-4.95e15 - 8.57e15i)T^{2} \)
19 \( 1 + (1.03e8 + 1.79e8i)T + (-2.10e16 + 3.64e16i)T^{2} \)
23 \( 1 + (1.66e8 + 2.89e8i)T + (-2.52e17 + 4.36e17i)T^{2} \)
29 \( 1 - 4.91e9T + 1.02e19T^{2} \)
31 \( 1 + (-2.19e9 + 3.80e9i)T + (-1.22e19 - 2.11e19i)T^{2} \)
37 \( 1 + (-4.86e9 - 8.43e9i)T + (-1.21e20 + 2.10e20i)T^{2} \)
41 \( 1 - 2.57e10T + 9.25e20T^{2} \)
43 \( 1 - 2.25e9T + 1.71e21T^{2} \)
47 \( 1 + (-6.30e10 - 1.09e11i)T + (-2.73e21 + 4.72e21i)T^{2} \)
53 \( 1 + (-8.56e10 + 1.48e11i)T + (-1.30e22 - 2.25e22i)T^{2} \)
59 \( 1 + (2.75e11 - 4.76e11i)T + (-5.24e22 - 9.09e22i)T^{2} \)
61 \( 1 + (2.20e10 + 3.82e10i)T + (-8.09e22 + 1.40e23i)T^{2} \)
67 \( 1 + (-6.07e10 + 1.05e11i)T + (-2.74e23 - 4.74e23i)T^{2} \)
71 \( 1 - 1.25e12T + 1.16e24T^{2} \)
73 \( 1 + (7.26e10 - 1.25e11i)T + (-8.35e23 - 1.44e24i)T^{2} \)
79 \( 1 + (1.26e12 + 2.19e12i)T + (-2.33e24 + 4.04e24i)T^{2} \)
83 \( 1 + 1.98e11T + 8.87e24T^{2} \)
89 \( 1 + (-2.74e12 - 4.74e12i)T + (-1.09e25 + 1.90e25i)T^{2} \)
97 \( 1 - 3.38e12T + 6.73e25T^{2} \)
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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

−17.08288319928730270832283063254, −15.34869298569985946635964656388, −14.28896486853434553949374505863, −13.31598700972583947009376192416, −10.88975057932151059442300315296, −10.05664053479435500006043773905, −7.49563047003995629097442026988, −6.36765821679055246556175587690, −4.49316695987840878654794507909, −2.43208766918718112815508495584, 0.792605522114908835596061840747, 2.10765151307443108673067106960, 4.73571821817269864954902540026, 5.91651713318611611728478057526, 8.690311953422511028809824857663, 9.770659498756858331779864675792, 12.06073893471133786420256492559, 12.62674698061786934872398576675, 13.97151181671692322653789999567, 15.78663239087316019810426442118

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