Properties

Label 2-14-7.3-c12-0-6
Degree $2$
Conductor $14$
Sign $-0.746 + 0.665i$
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 + (702. + 405. i)3-s + (−1.02e3 + 1.77e3i)4-s + (−1.83e4 + 1.05e4i)5-s − 3.67e4i·6-s + (−9.08e3 − 1.17e5i)7-s + 9.26e4·8-s + (6.33e4 + 1.09e5i)9-s + (8.28e5 + 4.78e5i)10-s + (6.51e5 − 1.12e6i)11-s + (−1.43e6 + 8.30e5i)12-s − 7.09e6i·13-s + (−4.39e6 + 3.01e6i)14-s − 1.71e7·15-s + (−2.09e6 − 3.63e6i)16-s + (−2.79e7 − 1.61e7i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.963 + 0.556i)3-s + (−0.249 + 0.433i)4-s + (−1.17 + 0.676i)5-s − 0.786i·6-s + (−0.0771 − 0.997i)7-s + 0.353·8-s + (0.119 + 0.206i)9-s + (0.828 + 0.478i)10-s + (0.367 − 0.637i)11-s + (−0.481 + 0.278i)12-s − 1.47i·13-s + (−0.583 + 0.399i)14-s − 1.50·15-s + (−0.125 − 0.216i)16-s + (−1.15 − 0.667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.746 + 0.665i$
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: \((2,\ 14,\ (\ :6),\ -0.746 + 0.665i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.294029 - 0.772232i\)
\(L(\frac12)\) \(\approx\) \(0.294029 - 0.772232i\)
\(L(7)\) 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 \( 1 + (9.08e3 + 1.17e5i)T \)
good3 \( 1 + (-702. - 405. i)T + (2.65e5 + 4.60e5i)T^{2} \)
5 \( 1 + (1.83e4 - 1.05e4i)T + (1.22e8 - 2.11e8i)T^{2} \)
11 \( 1 + (-6.51e5 + 1.12e6i)T + (-1.56e12 - 2.71e12i)T^{2} \)
13 \( 1 + 7.09e6iT - 2.32e13T^{2} \)
17 \( 1 + (2.79e7 + 1.61e7i)T + (2.91e14 + 5.04e14i)T^{2} \)
19 \( 1 + (2.32e7 - 1.34e7i)T + (1.10e15 - 1.91e15i)T^{2} \)
23 \( 1 + (-6.22e7 - 1.07e8i)T + (-1.09e16 + 1.89e16i)T^{2} \)
29 \( 1 + 7.52e8T + 3.53e17T^{2} \)
31 \( 1 + (-9.46e8 - 5.46e8i)T + (3.93e17 + 6.82e17i)T^{2} \)
37 \( 1 + (1.93e9 + 3.35e9i)T + (-3.29e18 + 5.70e18i)T^{2} \)
41 \( 1 - 2.55e9iT - 2.25e19T^{2} \)
43 \( 1 + 4.36e9T + 3.99e19T^{2} \)
47 \( 1 + (-8.04e9 + 4.64e9i)T + (5.80e19 - 1.00e20i)T^{2} \)
53 \( 1 + (1.09e8 - 1.89e8i)T + (-2.45e20 - 4.25e20i)T^{2} \)
59 \( 1 + (-5.45e10 - 3.14e10i)T + (8.89e20 + 1.54e21i)T^{2} \)
61 \( 1 + (-4.19e10 + 2.42e10i)T + (1.32e21 - 2.29e21i)T^{2} \)
67 \( 1 + (5.49e10 - 9.51e10i)T + (-4.09e21 - 7.08e21i)T^{2} \)
71 \( 1 + 7.43e10T + 1.64e22T^{2} \)
73 \( 1 + (2.26e11 + 1.30e11i)T + (1.14e22 + 1.98e22i)T^{2} \)
79 \( 1 + (1.20e11 + 2.08e11i)T + (-2.95e22 + 5.11e22i)T^{2} \)
83 \( 1 - 2.09e11iT - 1.06e23T^{2} \)
89 \( 1 + (-1.79e10 + 1.03e10i)T + (1.23e23 - 2.13e23i)T^{2} \)
97 \( 1 + 4.74e11iT - 6.93e23T^{2} \)
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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

−15.87632354683706912486370391392, −14.78770873792501532423039049139, −13.37601457121919113480339365287, −11.42269498375075557205602124167, −10.32821931307614101075441868515, −8.673700008837273974506970508577, −7.39826956894672433650206590422, −3.93685358828748657557797913547, −3.08199333162149872246663266904, −0.34720556549744322937305155915, 1.97719247753242695380527433964, 4.40688928631116044100001802856, 6.84684800176183052840160299598, 8.372178727014713255017645085331, 9.015129010988431488600735078470, 11.69091627144820164394242154442, 13.07745471850759651930485474262, 14.70693265644057973728512388218, 15.61082143515020366287550487545, 16.99609084053815757051386641670

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