↑

Properties

Label 2-208-13.10-c3-0-1
Degree $2$
Conductor $208$
Sign $-0.969 + 0.244i$
Analytic cond. $12.2723$
Root an. cond. $3.50319$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.93 + 6.82i)3-s + 3.30i·5-s + (−27.1 − 15.6i)7-s + (−17.5 + 30.3i)9-s + (−13.5 + 7.80i)11-s + (−45.2 − 12.2i)13-s + (−22.5 + 13.0i)15-s + (−53.8 + 93.2i)17-s + (−52.6 − 30.3i)19-s − 247. i·21-s + (62.1 + 107. i)23-s + 114.·25-s − 63.1·27-s + (−29.1 − 50.4i)29-s − 200. i·31-s + ⋯
L(s)  = 1  + (0.757 + 1.31i)3-s + 0.295i·5-s + (−1.46 − 0.847i)7-s + (−0.648 + 1.12i)9-s + (−0.370 + 0.213i)11-s + (−0.965 − 0.260i)13-s + (−0.388 + 0.224i)15-s + (−0.768 + 1.33i)17-s + (−0.635 − 0.366i)19-s − 2.56i·21-s + (0.563 + 0.975i)23-s + 0.912·25-s − 0.450·27-s + (−0.186 − 0.322i)29-s − 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.969 + 0.244i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ -0.969 + 0.244i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0897444 - 0.723280i\)
\(L(\frac12)\) \(\approx\) \(0.0897444 - 0.723280i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (45.2 + 12.2i)T \)
good3 \( 1 + (-3.93 - 6.82i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 3.30iT - 125T^{2} \)
7 \( 1 + (27.1 + 15.6i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (13.5 - 7.80i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (53.8 - 93.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (52.6 + 30.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-62.1 - 107. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (29.1 + 50.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 200. iT - 2.97e4T^{2} \)
37 \( 1 + (90.9 - 52.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (191. - 110. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (56.0 - 97.0i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 512. iT - 1.03e5T^{2} \)
53 \( 1 + 221.T + 1.48e5T^{2} \)
59 \( 1 + (-482. - 278. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-229. + 396. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (458. - 264. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (58.5 + 33.7i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 104. iT - 3.89e5T^{2} \)
79 \( 1 + 611.T + 4.93e5T^{2} \)
83 \( 1 + 491. iT - 5.71e5T^{2} \)
89 \( 1 + (-326. + 188. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (496. + 286. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−12.80832841091380663536293984200, −11.06154123510613456984578025075, −10.21741963404955988589748795851, −9.743443289014073330303036723605, −8.776258434106667523027460449538, −7.45627157620123267332983509891, −6.32953324782634800619961614038, −4.67914998947445039834648652266, −3.69786269118055832488179634267, −2.73494915407475065036542965430, 0.25815931323195849670102499316, 2.26220551047662465160038853065, 3.06653941414221395961692961985, 5.13461570862434628409550677319, 6.66046095799812253300603555428, 7.06666761305582519572723454692, 8.600355812553668683993872147606, 9.030205937935855774003543776958, 10.26936314057751238718607817329, 11.90583432977953414236193130686

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