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Properties

Label 2-208-13.12-c9-0-10
Degree $2$
Conductor $208$
Sign $-0.416 - 0.909i$
Analytic cond. $107.127$
Root an. cond. $10.3502$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 56.8·3-s + 1.60e3i·5-s − 9.63e3i·7-s − 1.64e4·9-s − 7.06e4i·11-s + (4.29e4 + 9.36e4i)13-s + 9.15e4i·15-s − 3.47e5·17-s − 7.08e5i·19-s − 5.47e5i·21-s − 6.36e5·23-s − 6.36e5·25-s − 2.05e6·27-s + 5.74e6·29-s + 6.21e6i·31-s + ⋯
L(s)  = 1  + 0.405·3-s + 1.15i·5-s − 1.51i·7-s − 0.835·9-s − 1.45i·11-s + (0.416 + 0.909i)13-s + 0.466i·15-s − 1.00·17-s − 1.24i·19-s − 0.614i·21-s − 0.474·23-s − 0.326·25-s − 0.744·27-s + 1.50·29-s + 1.20i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.416 - 0.909i$
Analytic conductor: \(107.127\)
Root analytic conductor: \(10.3502\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :9/2),\ -0.416 - 0.909i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.9461192639\)
\(L(\frac12)\) \(\approx\) \(0.9461192639\)
\(L(\frac{11}{2})\) 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 \( 1 + (-4.29e4 - 9.36e4i)T \)
good3 \( 1 - 56.8T + 1.96e4T^{2} \)
5 \( 1 - 1.60e3iT - 1.95e6T^{2} \)
7 \( 1 + 9.63e3iT - 4.03e7T^{2} \)
11 \( 1 + 7.06e4iT - 2.35e9T^{2} \)
17 \( 1 + 3.47e5T + 1.18e11T^{2} \)
19 \( 1 + 7.08e5iT - 3.22e11T^{2} \)
23 \( 1 + 6.36e5T + 1.80e12T^{2} \)
29 \( 1 - 5.74e6T + 1.45e13T^{2} \)
31 \( 1 - 6.21e6iT - 2.64e13T^{2} \)
37 \( 1 - 2.20e7iT - 1.29e14T^{2} \)
41 \( 1 - 8.84e6iT - 3.27e14T^{2} \)
43 \( 1 + 2.08e6T + 5.02e14T^{2} \)
47 \( 1 - 2.14e7iT - 1.11e15T^{2} \)
53 \( 1 - 3.39e7T + 3.29e15T^{2} \)
59 \( 1 + 9.00e7iT - 8.66e15T^{2} \)
61 \( 1 + 1.30e8T + 1.16e16T^{2} \)
67 \( 1 - 3.96e7iT - 2.72e16T^{2} \)
71 \( 1 - 1.32e8iT - 4.58e16T^{2} \)
73 \( 1 + 1.72e8iT - 5.88e16T^{2} \)
79 \( 1 + 3.61e8T + 1.19e17T^{2} \)
83 \( 1 - 3.15e8iT - 1.86e17T^{2} \)
89 \( 1 - 2.27e8iT - 3.50e17T^{2} \)
97 \( 1 - 1.23e9iT - 7.60e17T^{2} \)
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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

−11.03517114017336558553417353442, −10.33545072065164750162078951014, −8.952244734011915541251531387962, −8.134865413697051085477560722479, −6.80244049698077786607849178918, −6.41507428993667042640536292438, −4.60399148720734607978868181140, −3.41107504784631653170637550055, −2.72768746721380466174284203583, −1.03639684042559924054147288461, 0.20042267499872420625488146134, 1.80958520325215568331626875701, 2.61660275672371567633205318519, 4.17899579208728864138937057869, 5.32825672974182003993072306446, 6.03761510855634668464277796681, 7.76427067239389126102728429026, 8.646634840426238944392336522480, 9.113975130042431929353222433337, 10.24839997245575302923414753673

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