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Properties

Label 2-208-13.10-c3-0-6
Degree $2$
Conductor $208$
Sign $-0.169 - 0.985i$
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  + (1.93 + 3.35i)3-s + 13.8i·5-s + (14.2 + 8.24i)7-s + (5.98 − 10.3i)9-s + (54.5 − 31.4i)11-s + (−15.6 + 44.1i)13-s + (−46.3 + 26.7i)15-s + (−64.7 + 112. i)17-s + (−94.4 − 54.5i)19-s + 63.9i·21-s + (37.7 + 65.3i)23-s − 65.7·25-s + 151.·27-s + (−86.3 − 149. i)29-s + 188. i·31-s + ⋯
L(s)  = 1  + (0.373 + 0.646i)3-s + 1.23i·5-s + (0.770 + 0.445i)7-s + (0.221 − 0.383i)9-s + (1.49 − 0.862i)11-s + (−0.333 + 0.942i)13-s + (−0.798 + 0.460i)15-s + (−0.923 + 1.60i)17-s + (−1.14 − 0.658i)19-s + 0.664i·21-s + (0.342 + 0.592i)23-s − 0.525·25-s + 1.07·27-s + (−0.552 − 0.957i)29-s + 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.169 - 0.985i$
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: \((2,\ 208,\ (\ :3/2),\ -0.169 - 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.38988 + 1.64937i\)
\(L(\frac12)\) \(\approx\) \(1.38988 + 1.64937i\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.6 - 44.1i)T \)
good3 \( 1 + (-1.93 - 3.35i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 13.8iT - 125T^{2} \)
7 \( 1 + (-14.2 - 8.24i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-54.5 + 31.4i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (64.7 - 112. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (94.4 + 54.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-37.7 - 65.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (86.3 + 149. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 188. iT - 2.97e4T^{2} \)
37 \( 1 + (-124. + 71.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (241. - 139. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (27.0 - 46.8i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 280. iT - 1.03e5T^{2} \)
53 \( 1 - 550.T + 1.48e5T^{2} \)
59 \( 1 + (95.8 + 55.3i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (80.5 - 139. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-84.5 + 48.7i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-5.51 - 3.18i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 142. iT - 3.89e5T^{2} \)
79 \( 1 - 536.T + 4.93e5T^{2} \)
83 \( 1 + 1.32e3iT - 5.71e5T^{2} \)
89 \( 1 + (-927. + 535. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-963. - 556. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.85450808729503584871864421411, −11.20503074545221617397471956169, −10.35599469571817490435094584657, −9.099712643741479658925685058444, −8.557558045563969267476384934509, −6.85567447455168231945988097778, −6.25824224620411802099340567657, −4.38420014637167700689580437245, −3.52508711908419641653580618758, −1.94285837476742801100918729329, 0.953696753905712434118785769612, 2.10311806617830223064876143329, 4.30286471418698955225001986725, 5.01024775564244828865868060908, 6.76093391172608021821651793955, 7.67320674368404255396562220297, 8.608788122247293561833477133723, 9.454965904666031450127695533557, 10.75768525747052480425729143762, 11.93637349847412844870739128041

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