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Properties

Label 2-208-13.10-c3-0-13
Degree $2$
Conductor $208$
Sign $-0.0174 + 0.999i$
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  + (−2.43 − 4.22i)3-s − 0.110i·5-s + (14.0 + 8.09i)7-s + (1.61 − 2.79i)9-s + (33.6 − 19.4i)11-s + (−23.6 + 40.4i)13-s + (−0.464 + 0.268i)15-s + (47.2 − 81.7i)17-s + (−33.5 − 19.3i)19-s − 78.9i·21-s + (−46.2 − 80.0i)23-s + 124.·25-s − 147.·27-s + (−96.2 − 166. i)29-s − 158. i·31-s + ⋯
L(s)  = 1  + (−0.469 − 0.812i)3-s − 0.00985i·5-s + (0.757 + 0.437i)7-s + (0.0598 − 0.103i)9-s + (0.921 − 0.531i)11-s + (−0.503 + 0.863i)13-s + (−0.00800 + 0.00462i)15-s + (0.673 − 1.16i)17-s + (−0.404 − 0.233i)19-s − 0.820i·21-s + (−0.419 − 0.725i)23-s + 0.999·25-s − 1.05·27-s + (−0.616 − 1.06i)29-s − 0.918i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0174 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0174 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.0174 + 0.999i$
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: \((2,\ 208,\ (\ :3/2),\ -0.0174 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.08275 - 1.10180i\)
\(L(\frac12)\) \(\approx\) \(1.08275 - 1.10180i\)
\(L(\frac{5}{2})\) 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 \( 1 + (23.6 - 40.4i)T \)
good3 \( 1 + (2.43 + 4.22i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 0.110iT - 125T^{2} \)
7 \( 1 + (-14.0 - 8.09i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-33.6 + 19.4i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-47.2 + 81.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (33.5 + 19.3i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (46.2 + 80.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (96.2 + 166. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 158. iT - 2.97e4T^{2} \)
37 \( 1 + (248. - 143. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-202. + 117. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-262. + 455. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 320. iT - 1.03e5T^{2} \)
53 \( 1 - 414.T + 1.48e5T^{2} \)
59 \( 1 + (-223. - 128. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (71.6 - 124. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (392. - 226. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-654. - 377. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 641. iT - 3.89e5T^{2} \)
79 \( 1 - 588.T + 4.93e5T^{2} \)
83 \( 1 - 744. iT - 5.71e5T^{2} \)
89 \( 1 + (1.08e3 - 624. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.04e3 - 602. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.89451874148370625558650467534, −11.07249354676806327837375407154, −9.586727515992090901476200833096, −8.700003453267677819314383741591, −7.43474469378846753928329973710, −6.59538677642119468065436135825, −5.50717038114098358688100112434, −4.15079954731612439443456638747, −2.22120767141769997216332216079, −0.76766317383726213060003298622, 1.53020991392784857632542578449, 3.67316693149807028076287984017, 4.70980084537106366894207441714, 5.66641753383527784127514052298, 7.15221706349242935193617408453, 8.157927769601919692449547611275, 9.435861134795903471502108255908, 10.45606866102312893572902630778, 10.90272645113862036218408434426, 12.13095488997079562809328345160

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