Properties

Label 2-14-7.3-c12-0-7
Degree $2$
Conductor $14$
Sign $-0.995 + 0.0967i$
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  + (−22.6 − 39.1i)2-s + (−409. − 236. i)3-s + (−1.02e3 + 1.77e3i)4-s + (1.82e4 − 1.05e4i)5-s + 2.14e4i·6-s + (6.18e4 − 1.00e5i)7-s + 9.26e4·8-s + (−1.53e5 − 2.66e5i)9-s + (−8.24e5 − 4.76e5i)10-s + (−2.46e5 + 4.26e5i)11-s + (8.39e5 − 4.84e5i)12-s + 3.20e6i·13-s + (−5.32e6 − 1.59e5i)14-s − 9.95e6·15-s + (−2.09e6 − 3.63e6i)16-s + (−2.98e7 − 1.72e7i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.561 − 0.324i)3-s + (−0.249 + 0.433i)4-s + (1.16 − 0.673i)5-s + 0.458i·6-s + (0.525 − 0.850i)7-s + 0.353·8-s + (−0.289 − 0.501i)9-s + (−0.824 − 0.476i)10-s + (−0.139 + 0.241i)11-s + (0.280 − 0.162i)12-s + 0.664i·13-s + (−0.706 − 0.0211i)14-s − 0.873·15-s + (−0.125 − 0.216i)16-s + (−1.23 − 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0967i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.995 + 0.0967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.995 + 0.0967i$
Analytic conductor: \(12.7959\)
Root analytic conductor: \(3.57713\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :6),\ -0.995 + 0.0967i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0506290 - 1.04449i\)
\(L(\frac12)\) \(\approx\) \(0.0506290 - 1.04449i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (22.6 + 39.1i)T \)
7 \( 1 + (-6.18e4 + 1.00e5i)T \)
good3 \( 1 + (409. + 236. i)T + (2.65e5 + 4.60e5i)T^{2} \)
5 \( 1 + (-1.82e4 + 1.05e4i)T + (1.22e8 - 2.11e8i)T^{2} \)
11 \( 1 + (2.46e5 - 4.26e5i)T + (-1.56e12 - 2.71e12i)T^{2} \)
13 \( 1 - 3.20e6iT - 2.32e13T^{2} \)
17 \( 1 + (2.98e7 + 1.72e7i)T + (2.91e14 + 5.04e14i)T^{2} \)
19 \( 1 + (1.84e7 - 1.06e7i)T + (1.10e15 - 1.91e15i)T^{2} \)
23 \( 1 + (1.22e8 + 2.12e8i)T + (-1.09e16 + 1.89e16i)T^{2} \)
29 \( 1 - 8.71e8T + 3.53e17T^{2} \)
31 \( 1 + (6.16e8 + 3.55e8i)T + (3.93e17 + 6.82e17i)T^{2} \)
37 \( 1 + (-1.22e9 - 2.12e9i)T + (-3.29e18 + 5.70e18i)T^{2} \)
41 \( 1 + 4.29e9iT - 2.25e19T^{2} \)
43 \( 1 + 8.71e9T + 3.99e19T^{2} \)
47 \( 1 + (4.94e9 - 2.85e9i)T + (5.80e19 - 1.00e20i)T^{2} \)
53 \( 1 + (4.77e9 - 8.27e9i)T + (-2.45e20 - 4.25e20i)T^{2} \)
59 \( 1 + (-4.83e10 - 2.79e10i)T + (8.89e20 + 1.54e21i)T^{2} \)
61 \( 1 + (-5.97e10 + 3.45e10i)T + (1.32e21 - 2.29e21i)T^{2} \)
67 \( 1 + (-3.87e10 + 6.70e10i)T + (-4.09e21 - 7.08e21i)T^{2} \)
71 \( 1 + 1.07e11T + 1.64e22T^{2} \)
73 \( 1 + (-2.07e10 - 1.19e10i)T + (1.14e22 + 1.98e22i)T^{2} \)
79 \( 1 + (-1.59e10 - 2.76e10i)T + (-2.95e22 + 5.11e22i)T^{2} \)
83 \( 1 + 3.39e11iT - 1.06e23T^{2} \)
89 \( 1 + (-7.24e11 + 4.18e11i)T + (1.23e23 - 2.13e23i)T^{2} \)
97 \( 1 + 9.93e11iT - 6.93e23T^{2} \)
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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

−16.59475076900945527142062841853, −14.14956408739827453568603512367, −12.98257918363223760036827310333, −11.61092534918068529102865037677, −10.16037500454892821880553834588, −8.741713230174260934574569009838, −6.56847151531585022432319332078, −4.64479459626078561059966717626, −1.94296593655606891994358181238, −0.52305837950325399369436557306, 2.14905473816893497243667798972, 5.25799217370831938433320716950, 6.26450552740169547007947033696, 8.388847006008858239744615071910, 10.05025821676421541129418751655, 11.20382308605302345862733988826, 13.41535128707408234136890948727, 14.73476614219214980798343361834, 15.97109330627261157697652104154, 17.63141355665508704754758534023

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