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Properties

Label 2-208-13.3-c3-0-0
Degree $2$
Conductor $208$
Sign $-0.736 + 0.676i$
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  + (2.93 + 5.07i)3-s − 10.8·5-s + (−14.9 + 25.8i)7-s + (−3.70 + 6.41i)9-s + (−24.5 − 42.4i)11-s + (2.29 − 46.8i)13-s + (−31.8 − 55.1i)15-s + (3.03 − 5.26i)17-s + (−2.93 + 5.07i)19-s − 175.·21-s + (−2.79 − 4.84i)23-s − 6.94·25-s + 114.·27-s + (5.30 + 9.19i)29-s − 316.·31-s + ⋯
L(s)  = 1  + (0.564 + 0.977i)3-s − 0.971·5-s + (−0.806 + 1.39i)7-s + (−0.137 + 0.237i)9-s + (−0.672 − 1.16i)11-s + (0.0490 − 0.998i)13-s + (−0.548 − 0.950i)15-s + (0.0433 − 0.0750i)17-s + (−0.0354 + 0.0613i)19-s − 1.82·21-s + (−0.0253 − 0.0439i)23-s − 0.0555·25-s + 0.819·27-s + (0.0339 + 0.0588i)29-s − 1.83·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.736 + 0.676i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ -0.736 + 0.676i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0570206 - 0.146412i\)
\(L(\frac12)\) \(\approx\) \(0.0570206 - 0.146412i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.29 + 46.8i)T \)
good3 \( 1 + (-2.93 - 5.07i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 10.8T + 125T^{2} \)
7 \( 1 + (14.9 - 25.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (24.5 + 42.4i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-3.03 + 5.26i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.93 - 5.07i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (2.79 + 4.84i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-5.30 - 9.19i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 316.T + 2.97e4T^{2} \)
37 \( 1 + (-190. - 329. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (134. + 232. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (115. - 199. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 524.T + 1.03e5T^{2} \)
53 \( 1 + 274.T + 1.48e5T^{2} \)
59 \( 1 + (71.4 - 123. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (281. - 487. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (258. + 448. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (72.2 - 125. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 201.T + 3.89e5T^{2} \)
79 \( 1 + 26.4T + 4.93e5T^{2} \)
83 \( 1 - 1.14e3T + 5.71e5T^{2} \)
89 \( 1 + (-331. - 574. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-68.7 + 119. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.50128233834955585825451037016, −11.52725406186449214333461832743, −10.50805857671435046226316965501, −9.482070773910248365371954179763, −8.639780168120755856739700062843, −7.890434340383933443115854163642, −6.16740706399563449091641874721, −5.09749764498265035802144984623, −3.51849488502674899060589380952, −2.95967787016987925934862045657, 0.05955406927740758500350045395, 1.84565404563056820277632559402, 3.51889447788856861414274478832, 4.56394271717867037363321417756, 6.65649023655452299859157950647, 7.38514900242089200036069517870, 7.85249574633692292506947941022, 9.317066457484996449322383951925, 10.36681450767221897205270420031, 11.42825620350092801023616792337

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