Properties

Label 2-14-7.2-c9-0-1
Degree $2$
Conductor $14$
Sign $-0.0246 - 0.999i$
Analytic cond. $7.21050$
Root an. cond. $2.68523$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8 − 13.8i)2-s + (86.7 + 150. i)3-s + (−127. − 221. i)4-s + (−1.17e3 + 2.03e3i)5-s + 2.77e3·6-s + (−6.34e3 − 246. i)7-s − 4.09e3·8-s + (−5.20e3 + 9.01e3i)9-s + (1.87e4 + 3.25e4i)10-s + (3.55e4 + 6.15e4i)11-s + (2.22e4 − 3.84e4i)12-s + 8.94e4·13-s + (−5.41e4 + 8.59e4i)14-s − 4.06e5·15-s + (−3.27e4 + 5.67e4i)16-s + (−1.61e4 − 2.79e4i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.618 + 1.07i)3-s + (−0.249 − 0.433i)4-s + (−0.839 + 1.45i)5-s + 0.874·6-s + (−0.999 − 0.0387i)7-s − 0.353·8-s + (−0.264 + 0.457i)9-s + (0.593 + 1.02i)10-s + (0.732 + 1.26i)11-s + (0.309 − 0.535i)12-s + 0.868·13-s + (−0.377 + 0.598i)14-s − 2.07·15-s + (−0.125 + 0.216i)16-s + (−0.0468 − 0.0812i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0246 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0246 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.0246 - 0.999i$
Analytic conductor: \(7.21050\)
Root analytic conductor: \(2.68523\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :9/2),\ -0.0246 - 0.999i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.18555 + 1.21512i\)
\(L(\frac12)\) \(\approx\) \(1.18555 + 1.21512i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-8 + 13.8i)T \)
7 \( 1 + (6.34e3 + 246. i)T \)
good3 \( 1 + (-86.7 - 150. i)T + (-9.84e3 + 1.70e4i)T^{2} \)
5 \( 1 + (1.17e3 - 2.03e3i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (-3.55e4 - 6.15e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 - 8.94e4T + 1.06e10T^{2} \)
17 \( 1 + (1.61e4 + 2.79e4i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-3.89e5 + 6.74e5i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (8.86e5 - 1.53e6i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + 2.12e5T + 1.45e13T^{2} \)
31 \( 1 + (-2.15e6 - 3.73e6i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (3.10e6 - 5.37e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 6.40e6T + 3.27e14T^{2} \)
43 \( 1 + 6.99e6T + 5.02e14T^{2} \)
47 \( 1 + (-1.75e7 + 3.03e7i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (-2.13e7 - 3.69e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (-1.45e7 - 2.51e7i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (3.09e7 - 5.35e7i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (1.00e8 + 1.73e8i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 - 3.03e7T + 4.58e16T^{2} \)
73 \( 1 + (-1.45e8 - 2.52e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-5.44e7 + 9.42e7i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 - 2.16e8T + 1.86e17T^{2} \)
89 \( 1 + (-9.32e7 + 1.61e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 + 9.53e8T + 7.60e17T^{2} \)
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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

−18.11155513093879023743681790081, −15.70044176043329077880353555917, −15.16465242924347883440117381417, −13.84148501677535983820959804861, −11.84828559376463732881762898120, −10.44682153521670058716181759748, −9.384576392835783561870823139684, −6.89142093130583675919529851951, −4.00689270866333608640289995398, −3.06484796766458959250157072646, 0.811276141623970159815430949866, 3.74479079233610291051878332201, 6.14573766690199134306524848058, 7.967126662607652403091882262658, 8.851368649284368415064024725890, 12.05965097190921562683832018708, 12.99584086558268369700118339536, 13.99818466741991222204938684393, 16.02754227604660677413885046388, 16.58381326434673909243700963516

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