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Properties

Label 2-208-13.10-c3-0-10
Degree $2$
Conductor $208$
Sign $-0.564 - 0.825i$
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  + (4.48 + 7.76i)3-s + 11.8i·5-s + (27.8 + 16.0i)7-s + (−26.6 + 46.1i)9-s + (10.4 − 6.04i)11-s + (−4.16 − 46.6i)13-s + (−91.6 + 52.9i)15-s + (44.8 − 77.7i)17-s + (16.6 + 9.58i)19-s + 288. i·21-s + (−7.88 − 13.6i)23-s − 14.4·25-s − 235.·27-s + (−131. − 227. i)29-s − 84.0i·31-s + ⋯
L(s)  = 1  + (0.862 + 1.49i)3-s + 1.05i·5-s + (1.50 + 0.868i)7-s + (−0.987 + 1.71i)9-s + (0.286 − 0.165i)11-s + (−0.0888 − 0.996i)13-s + (−1.57 + 0.910i)15-s + (0.640 − 1.10i)17-s + (0.200 + 0.115i)19-s + 2.99i·21-s + (−0.0715 − 0.123i)23-s − 0.115·25-s − 1.68·27-s + (−0.840 − 1.45i)29-s − 0.486i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.564 - 0.825i$
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: \((2,\ 208,\ (\ :3/2),\ -0.564 - 0.825i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.25046 + 2.36991i\)
\(L(\frac12)\) \(\approx\) \(1.25046 + 2.36991i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.16 + 46.6i)T \)
good3 \( 1 + (-4.48 - 7.76i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 11.8iT - 125T^{2} \)
7 \( 1 + (-27.8 - 16.0i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-10.4 + 6.04i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-44.8 + 77.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-16.6 - 9.58i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (7.88 + 13.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (131. + 227. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 84.0iT - 2.97e4T^{2} \)
37 \( 1 + (234. - 135. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-143. + 82.6i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (135. - 234. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 238. iT - 1.03e5T^{2} \)
53 \( 1 + 94.2T + 1.48e5T^{2} \)
59 \( 1 + (-214. - 123. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (101. - 176. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-200. + 115. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (649. + 374. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 153. iT - 3.89e5T^{2} \)
79 \( 1 - 881.T + 4.93e5T^{2} \)
83 \( 1 - 197. iT - 5.71e5T^{2} \)
89 \( 1 + (-1.20e3 + 695. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (972. + 561. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.85898319379285346420738593099, −11.12758802973071791499348979920, −10.27624940498465231135390657062, −9.403738795226382966124549065812, −8.366850848967379001296023446903, −7.60309957256142527690937299804, −5.66192468805000765970922113140, −4.74131485290694425388894568392, −3.37151123932651122122777352195, −2.41160676820982834046638963000, 1.23351210668319516596639107811, 1.75116894806963877906961369489, 3.86556646300532651671054522956, 5.18101164248145105060722787713, 6.82498703405267475168008143708, 7.65771006629606164543189030251, 8.430605360931902199074414440088, 9.138179121827452939246701249173, 10.82002152616080946785400951752, 11.96318842781926254271500922777

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