Properties

Label 2-14-7.2-c13-0-2
Degree $2$
Conductor $14$
Sign $-0.610 + 0.791i$
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 + (1.14e3 + 1.97e3i)3-s + (−2.04e3 − 3.54e3i)4-s + (−1.82e4 + 3.16e4i)5-s − 1.46e5·6-s + (−2.33e5 − 2.05e5i)7-s + 2.62e5·8-s + (−1.81e6 + 3.13e6i)9-s + (−1.16e6 − 2.02e6i)10-s + (1.91e6 + 3.32e6i)11-s + (4.67e6 − 8.10e6i)12-s + 1.67e7·13-s + (1.88e7 − 6.39e6i)14-s − 8.34e7·15-s + (−8.38e6 + 1.45e7i)16-s + (−8.06e7 − 1.39e8i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.904 + 1.56i)3-s + (−0.249 − 0.433i)4-s + (−0.522 + 0.905i)5-s − 1.27·6-s + (−0.751 − 0.659i)7-s + 0.353·8-s + (−1.13 + 1.96i)9-s + (−0.369 − 0.639i)10-s + (0.326 + 0.565i)11-s + (0.452 − 0.783i)12-s + 0.961·13-s + (0.669 − 0.227i)14-s − 1.89·15-s + (−0.125 + 0.216i)16-s + (−0.810 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.610 + 0.791i$
Analytic conductor: \(15.0123\)
Root analytic conductor: \(3.87457\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :13/2),\ -0.610 + 0.791i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.489614 - 0.995838i\)
\(L(\frac12)\) \(\approx\) \(0.489614 - 0.995838i\)
\(L(\frac{15}{2})\) 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 \( 1 + (2.33e5 + 2.05e5i)T \)
good3 \( 1 + (-1.14e3 - 1.97e3i)T + (-7.97e5 + 1.38e6i)T^{2} \)
5 \( 1 + (1.82e4 - 3.16e4i)T + (-6.10e8 - 1.05e9i)T^{2} \)
11 \( 1 + (-1.91e6 - 3.32e6i)T + (-1.72e13 + 2.98e13i)T^{2} \)
13 \( 1 - 1.67e7T + 3.02e14T^{2} \)
17 \( 1 + (8.06e7 + 1.39e8i)T + (-4.95e15 + 8.57e15i)T^{2} \)
19 \( 1 + (1.29e8 - 2.24e8i)T + (-2.10e16 - 3.64e16i)T^{2} \)
23 \( 1 + (-4.39e8 + 7.61e8i)T + (-2.52e17 - 4.36e17i)T^{2} \)
29 \( 1 - 6.01e8T + 1.02e19T^{2} \)
31 \( 1 + (-1.66e9 - 2.88e9i)T + (-1.22e19 + 2.11e19i)T^{2} \)
37 \( 1 + (2.94e9 - 5.10e9i)T + (-1.21e20 - 2.10e20i)T^{2} \)
41 \( 1 + 4.54e10T + 9.25e20T^{2} \)
43 \( 1 + 2.12e10T + 1.71e21T^{2} \)
47 \( 1 + (3.51e10 - 6.09e10i)T + (-2.73e21 - 4.72e21i)T^{2} \)
53 \( 1 + (-1.57e10 - 2.72e10i)T + (-1.30e22 + 2.25e22i)T^{2} \)
59 \( 1 + (-1.02e11 - 1.77e11i)T + (-5.24e22 + 9.09e22i)T^{2} \)
61 \( 1 + (2.45e11 - 4.24e11i)T + (-8.09e22 - 1.40e23i)T^{2} \)
67 \( 1 + (9.70e9 + 1.68e10i)T + (-2.74e23 + 4.74e23i)T^{2} \)
71 \( 1 - 1.11e12T + 1.16e24T^{2} \)
73 \( 1 + (1.49e11 + 2.58e11i)T + (-8.35e23 + 1.44e24i)T^{2} \)
79 \( 1 + (1.60e12 - 2.77e12i)T + (-2.33e24 - 4.04e24i)T^{2} \)
83 \( 1 + 2.08e12T + 8.87e24T^{2} \)
89 \( 1 + (1.93e12 - 3.35e12i)T + (-1.09e25 - 1.90e25i)T^{2} \)
97 \( 1 - 1.42e13T + 6.73e25T^{2} \)
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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

−16.68069423160716123431052549587, −15.76537073250120435852916844930, −14.86004543853450148949062619330, −13.79317880345860236195186401500, −10.86282943091935312667125290279, −9.906757478871019293347248851753, −8.589159789185642137256894341914, −6.86164283814711361014969054967, −4.39681338548090048843719704097, −3.14885543660382024191752635163, 0.44854618777649585499647510590, 1.80194600749434751019646227686, 3.42693116673649309312866110859, 6.50638570272043917978182658438, 8.399545010182857562535603900483, 8.885010237846607070415632171893, 11.59473062211789240899649938567, 12.84743653767674607007475954990, 13.36774673388864449638158443359, 15.41495258950215157239085440355

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