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Properties

Label 2-14-7.6-c12-0-5
Degree $2$
Conductor $14$
Sign $-0.122 + 0.992i$
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  − 45.2·2-s − 102. i·3-s + 2.04e3·4-s − 6.91e3i·5-s + 4.61e3i·6-s + (−1.43e4 + 1.16e5i)7-s − 9.26e4·8-s + 5.21e5·9-s + 3.13e5i·10-s − 1.34e6·11-s − 2.08e5i·12-s − 5.40e6i·13-s + (6.50e5 − 5.28e6i)14-s − 7.05e5·15-s + 4.19e6·16-s − 4.33e7i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.139i·3-s + 0.500·4-s − 0.442i·5-s + 0.0989i·6-s + (−0.122 + 0.992i)7-s − 0.353·8-s + 0.980·9-s + 0.313i·10-s − 0.759·11-s − 0.0699i·12-s − 1.11i·13-s + (0.0864 − 0.701i)14-s − 0.0619·15-s + 0.250·16-s − 1.79i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(12.7959\)
Root analytic conductor: \(3.57713\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :6),\ -0.122 + 0.992i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.640788 - 0.724572i\)
\(L(\frac12)\) \(\approx\) \(0.640788 - 0.724572i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 45.2T \)
7 \( 1 + (1.43e4 - 1.16e5i)T \)
good3 \( 1 + 102. iT - 5.31e5T^{2} \)
5 \( 1 + 6.91e3iT - 2.44e8T^{2} \)
11 \( 1 + 1.34e6T + 3.13e12T^{2} \)
13 \( 1 + 5.40e6iT - 2.32e13T^{2} \)
17 \( 1 + 4.33e7iT - 5.82e14T^{2} \)
19 \( 1 + 4.81e7iT - 2.21e15T^{2} \)
23 \( 1 + 1.61e8T + 2.19e16T^{2} \)
29 \( 1 + 7.77e7T + 3.53e17T^{2} \)
31 \( 1 + 1.36e8iT - 7.87e17T^{2} \)
37 \( 1 + 1.35e9T + 6.58e18T^{2} \)
41 \( 1 + 4.10e9iT - 2.25e19T^{2} \)
43 \( 1 - 2.00e9T + 3.99e19T^{2} \)
47 \( 1 + 1.64e10iT - 1.16e20T^{2} \)
53 \( 1 - 2.76e10T + 4.91e20T^{2} \)
59 \( 1 - 3.80e10iT - 1.77e21T^{2} \)
61 \( 1 - 8.36e10iT - 2.65e21T^{2} \)
67 \( 1 - 9.92e10T + 8.18e21T^{2} \)
71 \( 1 + 3.72e10T + 1.64e22T^{2} \)
73 \( 1 + 7.69e10iT - 2.29e22T^{2} \)
79 \( 1 + 2.27e11T + 5.90e22T^{2} \)
83 \( 1 + 1.32e11iT - 1.06e23T^{2} \)
89 \( 1 + 5.07e11iT - 2.46e23T^{2} \)
97 \( 1 - 2.74e11iT - 6.93e23T^{2} \)
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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

−16.11747395728818622152983958288, −15.40192362841605313559663652196, −13.19833960305388638936738761849, −11.96624377323754922043034793593, −10.18982763332179002531822908097, −8.829918477932036345739879649089, −7.30488118760261813420556369246, −5.27767147032163271855782757251, −2.53022319685075672309084272612, −0.53178389396024155444550900769, 1.60508614472873612325750055259, 3.96185746391561968547740601369, 6.57767940713249198214778754535, 7.946074885067406883824150080580, 9.937348668520805164462364796461, 10.78041354884955175704295839773, 12.71266369181641261993880638676, 14.37862993297035784282976955726, 15.90272036876835378890243672873, 16.96844735316577353704532874110

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