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Properties

Label 2-208-13.2-c2-0-6
Degree $2$
Conductor $208$
Sign $0.749 + 0.661i$
Analytic cond. $5.66758$
Root an. cond. $2.38066$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.12 − 1.95i)3-s + (−6.17 + 6.17i)5-s + (8.43 − 2.26i)7-s + (1.95 − 3.39i)9-s + (1.62 − 6.05i)11-s + (12.2 − 4.45i)13-s + (19.0 + 5.09i)15-s + (16.1 + 9.29i)17-s + (−6.58 − 24.5i)19-s + (−13.9 − 13.9i)21-s + (17.9 − 10.3i)23-s − 51.3i·25-s − 29.1·27-s + (17.0 + 29.5i)29-s + (−3.58 + 3.58i)31-s + ⋯
L(s)  = 1  + (−0.375 − 0.650i)3-s + (−1.23 + 1.23i)5-s + (1.20 − 0.322i)7-s + (0.217 − 0.376i)9-s + (0.147 − 0.550i)11-s + (0.939 − 0.342i)13-s + (1.26 + 0.339i)15-s + (0.947 + 0.546i)17-s + (−0.346 − 1.29i)19-s + (−0.662 − 0.662i)21-s + (0.780 − 0.450i)23-s − 2.05i·25-s − 1.07·27-s + (0.588 + 1.01i)29-s + (−0.115 + 0.115i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(5.66758\)
Root analytic conductor: \(2.38066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1),\ 0.749 + 0.661i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.21295 - 0.458819i\)
\(L(\frac12)\) \(\approx\) \(1.21295 - 0.458819i\)
\(L(2)\) 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 \( 1 + (-12.2 + 4.45i)T \)
good3 \( 1 + (1.12 + 1.95i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (6.17 - 6.17i)T - 25iT^{2} \)
7 \( 1 + (-8.43 + 2.26i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.62 + 6.05i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-16.1 - 9.29i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (6.58 + 24.5i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (-17.9 + 10.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-17.0 - 29.5i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (3.58 - 3.58i)T - 961iT^{2} \)
37 \( 1 + (-10.2 + 38.4i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-1.97 - 0.529i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (-67.2 - 38.8i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-4.36 - 4.36i)T + 2.20e3iT^{2} \)
53 \( 1 - 14.4T + 2.80e3T^{2} \)
59 \( 1 + (9.29 - 2.49i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (35.2 - 61.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (80.4 + 21.5i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (20.3 + 75.9i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (23.3 + 23.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 26.9T + 6.24e3T^{2} \)
83 \( 1 + (69.3 - 69.3i)T - 6.88e3iT^{2} \)
89 \( 1 + (0.856 - 3.19i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (9.26 + 34.5i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−11.87718478806725617173234091963, −10.94286309045321467689657546819, −10.80193058898415822149224334916, −8.776292077704467742949438370255, −7.75020542859182357259494255463, −7.09057717532817608280245535922, −6.04255179392355559192325062333, −4.32980157831600290765255339592, −3.15049526973385212315323995274, −0.969139687283565111152911110403, 1.36768558796242568407812608020, 3.97324469181692723836715349659, 4.67754556954815036314720791582, 5.57876143814108669808456411063, 7.60769248838034175489838515341, 8.198726705128751396712866639685, 9.206184943661739371704481299229, 10.45660489160831760515288384754, 11.56433802131466611655748024473, 11.92087367663491157176338493850

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