Properties

Label 2-208-13.12-c5-0-13
Degree $2$
Conductor $208$
Sign $0.832 - 0.554i$
Analytic cond. $33.3598$
Root an. cond. $5.77579$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 68i·5-s − 82i·7-s − 227·9-s − 390i·11-s + (507 − 338i)13-s − 272i·15-s + 1.73e3·17-s + 1.07e3i·19-s + 328i·21-s − 2.10e3·23-s − 1.49e3·25-s + 1.88e3·27-s − 1.69e3·29-s + 1.43e3i·31-s + ⋯
L(s)  = 1  − 0.256·3-s + 1.21i·5-s − 0.632i·7-s − 0.934·9-s − 0.971i·11-s + (0.832 − 0.554i)13-s − 0.312i·15-s + 1.45·17-s + 0.682i·19-s + 0.162i·21-s − 0.829·23-s − 0.479·25-s + 0.496·27-s − 0.373·29-s + 0.267i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.832 - 0.554i$
Analytic conductor: \(33.3598\)
Root analytic conductor: \(5.77579\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :5/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.660867438\)
\(L(\frac12)\) \(\approx\) \(1.660867438\)
\(L(\frac{7}{2})\) 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 \( 1 + (-507 + 338i)T \)
good3 \( 1 + 4T + 243T^{2} \)
5 \( 1 - 68iT - 3.12e3T^{2} \)
7 \( 1 + 82iT - 1.68e4T^{2} \)
11 \( 1 + 390iT - 1.61e5T^{2} \)
17 \( 1 - 1.73e3T + 1.41e6T^{2} \)
19 \( 1 - 1.07e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.10e3T + 6.43e6T^{2} \)
29 \( 1 + 1.69e3T + 2.05e7T^{2} \)
31 \( 1 - 1.43e3iT - 2.86e7T^{2} \)
37 \( 1 - 8.85e3iT - 6.93e7T^{2} \)
41 \( 1 - 6.76e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.69e4T + 1.47e8T^{2} \)
47 \( 1 - 2.51e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.82e4T + 4.18e8T^{2} \)
59 \( 1 + 2.12e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.45e3T + 8.44e8T^{2} \)
67 \( 1 + 4.45e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.77e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.10e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.53e4T + 3.07e9T^{2} \)
83 \( 1 - 1.24e5iT - 3.93e9T^{2} \)
89 \( 1 + 1.87e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.21e5iT - 8.58e9T^{2} \)
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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

−11.32130246305943240617634152246, −10.76386582508757035017177866402, −9.945044199774557823258956795370, −8.410202676992135155441958743039, −7.59912852813331805358807038967, −6.26217517624705967393944835008, −5.65519726705738802948665903337, −3.69233389502079719821369663458, −2.92273571815553403742346210219, −0.902808299263636358242845443181, 0.72391915173164107180634104818, 2.18555321159348322511152676718, 3.94893951499066450441792125206, 5.23758935001953228120508882866, 5.90580935836778145117691479872, 7.46405586537184223929370653119, 8.681966816636957201950579041419, 9.171316989696736994436351324465, 10.43062174852545905451908578722, 11.80430357542368049771325009319

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