Properties

Label 2-14-7.6-c6-0-3
Degree $2$
Conductor $14$
Sign $0.570 + 0.821i$
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 − 29.9i·3-s + 32.0·4-s − 98.3i·5-s − 169. i·6-s + (195. + 281. i)7-s + 181.·8-s − 168.·9-s − 556. i·10-s − 1.04e3·11-s − 958. i·12-s + 3.66e3i·13-s + (1.10e3 + 1.59e3i)14-s − 2.94e3·15-s + 1.02e3·16-s + 1.85e3i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.10i·3-s + 0.500·4-s − 0.787i·5-s − 0.784i·6-s + (0.570 + 0.821i)7-s + 0.353·8-s − 0.230·9-s − 0.556i·10-s − 0.782·11-s − 0.554i·12-s + 1.66i·13-s + (0.403 + 0.580i)14-s − 0.873·15-s + 0.250·16-s + 0.377i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14\)    =    \(2 \cdot 7\)
Sign: $0.570 + 0.821i$
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: \((2,\ 14,\ (\ :3),\ 0.570 + 0.821i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.82397 - 0.953390i\)
\(L(\frac12)\) \(\approx\) \(1.82397 - 0.953390i\)
\(L(4)\) 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 \( 1 + (-195. - 281. i)T \)
good3 \( 1 + 29.9iT - 729T^{2} \)
5 \( 1 + 98.3iT - 1.56e4T^{2} \)
11 \( 1 + 1.04e3T + 1.77e6T^{2} \)
13 \( 1 - 3.66e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.85e3iT - 2.41e7T^{2} \)
19 \( 1 + 7.49e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.16e4T + 1.48e8T^{2} \)
29 \( 1 + 4.16e4T + 5.94e8T^{2} \)
31 \( 1 - 2.20e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.91e4T + 2.56e9T^{2} \)
41 \( 1 - 1.80e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.14e5T + 6.32e9T^{2} \)
47 \( 1 + 1.39e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.24e5T + 2.21e10T^{2} \)
59 \( 1 + 2.71e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.38e5iT - 5.15e10T^{2} \)
67 \( 1 + 1.06e5T + 9.04e10T^{2} \)
71 \( 1 - 2.16e5T + 1.28e11T^{2} \)
73 \( 1 + 1.11e5iT - 1.51e11T^{2} \)
79 \( 1 - 4.51e5T + 2.43e11T^{2} \)
83 \( 1 - 2.25e5iT - 3.26e11T^{2} \)
89 \( 1 - 4.11e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.17e5iT - 8.32e11T^{2} \)
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

−18.26584224930938946690351683233, −16.69073178740801040981771543129, −15.16277716787190170424509629038, −13.56451171486002962272060594041, −12.61267282436283628360058488894, −11.47619625570003954078704736690, −8.743131243383779283035843669582, −6.97470933455247246630931941210, −5.04012306530305913138641290254, −1.86371346581523819161058772347, 3.44123114552857990266343436475, 5.21015436235711202081742289833, 7.56705771619632141047465505399, 10.24306775010029967900648671490, 10.94433243691035871113541021821, 13.09515784612772290387638888658, 14.65785795898302620700388099959, 15.41895143814995947830683683374, 16.83510291101240944324770906585, 18.44916164614678172508165051739

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