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Properties

Label 2-208-13.3-c3-0-3
Degree $2$
Conductor $208$
Sign $0.286 - 0.957i$
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  + (−1.84 − 3.19i)3-s − 17.8·5-s + (2.71 − 4.70i)7-s + (6.71 − 11.6i)9-s + (−11.2 − 19.4i)11-s + (21.9 + 41.4i)13-s + (32.8 + 56.8i)15-s + (−33.9 + 58.8i)17-s + (−40.4 + 69.9i)19-s − 20.0·21-s + (70.2 + 121. i)23-s + 192.·25-s − 148.·27-s + (53.3 + 92.3i)29-s + 276.·31-s + ⋯
L(s)  = 1  + (−0.354 − 0.614i)3-s − 1.59·5-s + (0.146 − 0.254i)7-s + (0.248 − 0.430i)9-s + (−0.307 − 0.532i)11-s + (0.468 + 0.883i)13-s + (0.564 + 0.978i)15-s + (−0.484 + 0.839i)17-s + (−0.487 + 0.844i)19-s − 0.208·21-s + (0.637 + 1.10i)23-s + 1.53·25-s − 1.06·27-s + (0.341 + 0.591i)29-s + 1.59·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.286 - 0.957i$
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: \((2,\ 208,\ (\ :3/2),\ 0.286 - 0.957i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.483112 + 0.359595i\)
\(L(\frac12)\) \(\approx\) \(0.483112 + 0.359595i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-21.9 - 41.4i)T \)
good3 \( 1 + (1.84 + 3.19i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 17.8T + 125T^{2} \)
7 \( 1 + (-2.71 + 4.70i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (11.2 + 19.4i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (33.9 - 58.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (40.4 - 69.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-70.2 - 121. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-53.3 - 92.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 276.T + 2.97e4T^{2} \)
37 \( 1 + (-2.14 - 3.71i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (113. + 197. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-13.7 + 23.8i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 318.T + 1.03e5T^{2} \)
53 \( 1 + 67.6T + 1.48e5T^{2} \)
59 \( 1 + (145. - 252. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (331. - 574. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (212. + 368. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (76.4 - 132. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 117.T + 3.89e5T^{2} \)
79 \( 1 + 202.T + 4.93e5T^{2} \)
83 \( 1 + 336.T + 5.71e5T^{2} \)
89 \( 1 + (359. + 621. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (379. - 657. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.99588537873179174965680537628, −11.42084792030104580360149024383, −10.48815045293813200259029326362, −8.882340283054720313351890822016, −8.007584292587116234865897137639, −7.09721349400526444211853586412, −6.14196051986409894309697857396, −4.41077690614290749215004998711, −3.50682921692832501213890944072, −1.24125589720344079733721313808, 0.30834867005628131004870103814, 2.86630948201294585974339306291, 4.39272436796985208093542036400, 4.92998826249953608480353092086, 6.69575007228311692564000449169, 7.82369163724607151557399181411, 8.557476418477847054978989887713, 9.983500404567403976971943204887, 10.93859708443139167939184088721, 11.53063303355036817259986009677

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