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Properties

Label 2-208-13.12-c11-0-31
Degree $2$
Conductor $208$
Sign $0.999 + 0.0120i$
Analytic cond. $159.815$
Root an. cond. $12.6418$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 143.·3-s + 5.26e3i·5-s − 1.61e4i·7-s − 1.56e5·9-s + 4.47e5i·11-s + (−1.33e6 − 1.61e4i)13-s + 7.54e5i·15-s − 6.84e6·17-s − 1.68e7i·19-s − 2.31e6i·21-s − 1.49e7·23-s + 2.10e7·25-s − 4.78e7·27-s + 3.19e7·29-s − 2.68e7i·31-s + ⋯
L(s)  = 1  + 0.340·3-s + 0.753i·5-s − 0.364i·7-s − 0.884·9-s + 0.837i·11-s + (−0.999 − 0.0120i)13-s + 0.256i·15-s − 1.16·17-s − 1.56i·19-s − 0.123i·21-s − 0.483·23-s + 0.432·25-s − 0.641·27-s + 0.289·29-s − 0.168i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0120i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.999 + 0.0120i$
Analytic conductor: \(159.815\)
Root analytic conductor: \(12.6418\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :11/2),\ 0.999 + 0.0120i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.486766811\)
\(L(\frac12)\) \(\approx\) \(1.486766811\)
\(L(\frac{13}{2})\) 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 \( 1 + (1.33e6 + 1.61e4i)T \)
good3 \( 1 - 143.T + 1.77e5T^{2} \)
5 \( 1 - 5.26e3iT - 4.88e7T^{2} \)
7 \( 1 + 1.61e4iT - 1.97e9T^{2} \)
11 \( 1 - 4.47e5iT - 2.85e11T^{2} \)
17 \( 1 + 6.84e6T + 3.42e13T^{2} \)
19 \( 1 + 1.68e7iT - 1.16e14T^{2} \)
23 \( 1 + 1.49e7T + 9.52e14T^{2} \)
29 \( 1 - 3.19e7T + 1.22e16T^{2} \)
31 \( 1 + 2.68e7iT - 2.54e16T^{2} \)
37 \( 1 - 5.43e8iT - 1.77e17T^{2} \)
41 \( 1 + 9.19e8iT - 5.50e17T^{2} \)
43 \( 1 - 2.47e8T + 9.29e17T^{2} \)
47 \( 1 + 1.05e9iT - 2.47e18T^{2} \)
53 \( 1 - 3.13e9T + 9.26e18T^{2} \)
59 \( 1 - 2.01e9iT - 3.01e19T^{2} \)
61 \( 1 - 1.47e9T + 4.35e19T^{2} \)
67 \( 1 + 1.34e10iT - 1.22e20T^{2} \)
71 \( 1 - 2.43e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.78e10iT - 3.13e20T^{2} \)
79 \( 1 + 1.03e10T + 7.47e20T^{2} \)
83 \( 1 - 3.76e10iT - 1.28e21T^{2} \)
89 \( 1 - 9.91e10iT - 2.77e21T^{2} \)
97 \( 1 + 4.38e10iT - 7.15e21T^{2} \)
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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

−10.43083665905449439039159245715, −9.418516329761088587405001418848, −8.485343286627340870434089130923, −7.22092395592796380539543201437, −6.70479275284928415808011140142, −5.20782785107597378898660692298, −4.17169919792538301913973988613, −2.77361907973075772743980447869, −2.24338567477648227669263712765, −0.44635923297323275714441791052, 0.55829878897004531873240587455, 1.95516733904526264102300638715, 2.95116632704683007278260626837, 4.20751020952690878678596541203, 5.36961274836334645035931151282, 6.18986361529706695737227950947, 7.68263837236417033924203265387, 8.585464575869682070918277312172, 9.143018728293711121542613940433, 10.35278953105686165195441920285

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