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Properties

Label 2-14-7.3-c4-0-1
Degree $2$
Conductor $14$
Sign $-0.701 + 0.712i$
Analytic cond. $1.44717$
Root an. cond. $1.20298$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 2.44i)2-s + (−12.9 − 7.49i)3-s + (−3.99 + 6.92i)4-s + (21.9 − 12.6i)5-s + 42.4i·6-s + (−7 − 48.4i)7-s + 22.6·8-s + (71.9 + 124. i)9-s + (−62.1 − 35.9i)10-s + (−21.9 + 38.0i)11-s + (103. − 59.9i)12-s − 162. i·13-s + (−108. + 85.7i)14-s − 380.·15-s + (−32.0 − 55.4i)16-s + (93.7 + 54.1i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−1.44 − 0.833i)3-s + (−0.249 + 0.433i)4-s + (0.879 − 0.507i)5-s + 1.17i·6-s + (−0.142 − 0.989i)7-s + 0.353·8-s + (0.887 + 1.53i)9-s + (−0.621 − 0.359i)10-s + (−0.181 + 0.314i)11-s + (0.721 − 0.416i)12-s − 0.961i·13-s + (−0.555 + 0.437i)14-s − 1.69·15-s + (−0.125 − 0.216i)16-s + (0.324 + 0.187i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(1.44717\)
Root analytic conductor: \(1.20298\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :2),\ -0.701 + 0.712i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.253593 - 0.605115i\)
\(L(\frac12)\) \(\approx\) \(0.253593 - 0.605115i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 2.44i)T \)
7 \( 1 + (7 + 48.4i)T \)
good3 \( 1 + (12.9 + 7.49i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (-21.9 + 12.6i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (21.9 - 38.0i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 162. iT - 2.85e4T^{2} \)
17 \( 1 + (-93.7 - 54.1i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-389. + 224. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-217. - 377. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 - 742.T + 7.07e5T^{2} \)
31 \( 1 + (900. + 519. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (493. + 854. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 1.14e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.41e3T + 3.41e6T^{2} \)
47 \( 1 + (1.16e3 - 672. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-1.78e3 + 3.08e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-3.71e3 - 2.14e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.20e3 - 698. i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.63e3 - 6.29e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 5.98e3T + 2.54e7T^{2} \)
73 \( 1 + (500. + 288. i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-1.57e3 - 2.72e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 4.72e3iT - 4.74e7T^{2} \)
89 \( 1 + (725. - 418. i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 5.62e3iT - 8.85e7T^{2} \)
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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

−17.90493428600633552732742037871, −17.51964687604231379837299351443, −16.40132733082564426682676400971, −13.47264567523626169559698605848, −12.67222590269912192973664680883, −11.17680176016329890874710793722, −9.915111920181594314695337083319, −7.34983296192682127925066568101, −5.42076147750614845810174142746, −0.999557796095517861937555837908, 5.32191428348983662838364376165, 6.41554897790853692003439734286, 9.331726816700467104543229582288, 10.51937692744744366146364942026, 12.00328351079658417387847940792, 14.24931730731304178026717873469, 15.78398286683580360403139004012, 16.64055925748531110228505528959, 17.87540552180749074362321979739, 18.67480006959931329427516265674

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