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Properties

Label 2-208-13.10-c3-0-9
Degree $2$
Conductor $208$
Sign $0.969 + 0.246i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.230 − 0.398i)3-s + 11.9i·5-s + (−26.4 − 15.2i)7-s + (13.3 − 23.1i)9-s + (36.8 − 21.2i)11-s + (33.9 + 32.2i)13-s + (4.77 − 2.75i)15-s + (26.3 − 45.6i)17-s + (111. + 64.5i)19-s + 14.0i·21-s + (54.2 + 93.9i)23-s − 18.5·25-s − 24.7·27-s + (−44.1 − 76.5i)29-s − 32.1i·31-s + ⋯
L(s)  = 1  + (−0.0443 − 0.0767i)3-s + 1.07i·5-s + (−1.42 − 0.824i)7-s + (0.496 − 0.859i)9-s + (1.00 − 0.582i)11-s + (0.724 + 0.688i)13-s + (0.0822 − 0.0475i)15-s + (0.375 − 0.650i)17-s + (1.34 + 0.778i)19-s + 0.146i·21-s + (0.491 + 0.852i)23-s − 0.148·25-s − 0.176·27-s + (−0.282 − 0.489i)29-s − 0.186i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.969 + 0.246i$
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: \((2,\ 208,\ (\ :3/2),\ 0.969 + 0.246i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.65272 - 0.207007i\)
\(L(\frac12)\) \(\approx\) \(1.65272 - 0.207007i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-33.9 - 32.2i)T \)
good3 \( 1 + (0.230 + 0.398i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 11.9iT - 125T^{2} \)
7 \( 1 + (26.4 + 15.2i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-36.8 + 21.2i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-26.3 + 45.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-111. - 64.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-54.2 - 93.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (44.1 + 76.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 32.1iT - 2.97e4T^{2} \)
37 \( 1 + (-300. + 173. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-157. + 90.9i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (37.8 - 65.5i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 307. iT - 1.03e5T^{2} \)
53 \( 1 + 693.T + 1.48e5T^{2} \)
59 \( 1 + (314. + 181. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-214. + 371. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-169. + 98.1i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-907. - 523. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 234. iT - 3.89e5T^{2} \)
79 \( 1 + 409.T + 4.93e5T^{2} \)
83 \( 1 - 13.1iT - 5.71e5T^{2} \)
89 \( 1 + (-552. + 319. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.06e3 + 612. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.76755453914912948379287681619, −11.00175033885245252519332817611, −9.660876779576428670696887349570, −9.441421311910826988297272765507, −7.49044238882432036633904164452, −6.70287817091393450202993427209, −6.06598154002469080533635380609, −3.77303539958083690590864569837, −3.31714358629335215137300058145, −0.944444297645829075868440817784, 1.16463533527834403520848195022, 3.03917293735205067185704385152, 4.52534230869626352594552367796, 5.64493346241402054605707473198, 6.73572462318663202564095634846, 8.109970876244888281231354456682, 9.198758898219803207809179905455, 9.715406186081150312151364251052, 11.01266951469285672495025256822, 12.33460534874160371812932980183

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