Properties

Label 2-14-7.6-c6-0-0
Degree $2$
Conductor $14$
Sign $0.121 - 0.992i$
Analytic cond. $3.22075$
Root an. cond. $1.79464$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.65·2-s − 3.84i·3-s + 32.0·4-s + 204. i·5-s + 21.7i·6-s + (−41.7 + 340. i)7-s − 181.·8-s + 714.·9-s − 1.15e3i·10-s − 1.17e3·11-s − 122. i·12-s − 1.39e3i·13-s + (236. − 1.92e3i)14-s + 786.·15-s + 1.02e3·16-s + 6.06e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.142i·3-s + 0.500·4-s + 1.63i·5-s + 0.100i·6-s + (−0.121 + 0.992i)7-s − 0.353·8-s + 0.979·9-s − 1.15i·10-s − 0.884·11-s − 0.0711i·12-s − 0.637i·13-s + (0.0861 − 0.701i)14-s + 0.233·15-s + 0.250·16-s + 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.121 - 0.992i$
Analytic conductor: \(3.22075\)
Root analytic conductor: \(1.79464\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{14} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 14,\ (\ :3),\ 0.121 - 0.992i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.712827 + 0.630670i\)
\(L(\frac12)\) \(\approx\) \(0.712827 + 0.630670i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65T \)
7 \( 1 + (41.7 - 340. i)T \)
good3 \( 1 + 3.84iT - 729T^{2} \)
5 \( 1 - 204. iT - 1.56e4T^{2} \)
11 \( 1 + 1.17e3T + 1.77e6T^{2} \)
13 \( 1 + 1.39e3iT - 4.82e6T^{2} \)
17 \( 1 - 6.06e3iT - 2.41e7T^{2} \)
19 \( 1 + 3.13e3iT - 4.70e7T^{2} \)
23 \( 1 - 8.65e3T + 1.48e8T^{2} \)
29 \( 1 - 3.25e4T + 5.94e8T^{2} \)
31 \( 1 + 4.22e4iT - 8.87e8T^{2} \)
37 \( 1 - 3.75e4T + 2.56e9T^{2} \)
41 \( 1 - 9.35e4iT - 4.75e9T^{2} \)
43 \( 1 - 9.20e4T + 6.32e9T^{2} \)
47 \( 1 - 2.49e4iT - 1.07e10T^{2} \)
53 \( 1 - 109.T + 2.21e10T^{2} \)
59 \( 1 + 3.26e5iT - 4.21e10T^{2} \)
61 \( 1 + 8.45e4iT - 5.15e10T^{2} \)
67 \( 1 + 1.11e5T + 9.04e10T^{2} \)
71 \( 1 + 4.41e5T + 1.28e11T^{2} \)
73 \( 1 - 4.64e5iT - 1.51e11T^{2} \)
79 \( 1 - 5.95e5T + 2.43e11T^{2} \)
83 \( 1 - 6.54e5iT - 3.26e11T^{2} \)
89 \( 1 + 2.98e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.69e5iT - 8.32e11T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.56442240284074113158727856741, −17.76389103660354726147601050344, −15.64724378542896190444631665480, −14.95232999854192493566121403808, −12.85915816698137519740319576869, −11.04606202617613967104475756779, −9.954250041488454821248368793856, −7.85411874414407392759396515208, −6.33859245696523188220522060595, −2.65164762697882211504464742228, 0.961920847447946679456680070997, 4.69914401864769701379118414498, 7.39285511371144148459613847782, 9.023600513172322870039378606323, 10.34238995854355790171013150365, 12.30240115825826189790306003323, 13.57004555493516136898908971881, 15.96364077901677901272266056601, 16.46931786760479609090359133281, 17.84229163890315113018370434649

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