Properties

Label 2-208-13.10-c3-0-19
Degree $2$
Conductor $208$
Sign $-0.711 - 0.702i$
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  + (−3.5 − 6.06i)3-s − 13.8i·5-s + (−19.5 − 11.2i)7-s + (−11 + 19.0i)9-s + (19.5 − 11.2i)11-s + (−13 − 45.0i)13-s + (−84 + 48.4i)15-s + (−13.5 + 23.3i)17-s + (76.5 + 44.1i)19-s + 157. i·21-s + (28.5 + 49.3i)23-s − 66.9·25-s − 35.0·27-s + (34.5 + 59.7i)29-s − 72.7i·31-s + ⋯
L(s)  = 1  + (−0.673 − 1.16i)3-s − 1.23i·5-s + (−1.05 − 0.607i)7-s + (−0.407 + 0.705i)9-s + (0.534 − 0.308i)11-s + (−0.277 − 0.960i)13-s + (−1.44 + 0.834i)15-s + (−0.192 + 0.333i)17-s + (0.923 + 0.533i)19-s + 1.63i·21-s + (0.258 + 0.447i)23-s − 0.535·25-s − 0.249·27-s + (0.220 + 0.382i)29-s − 0.421i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.711 - 0.702i$
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: \((2,\ 208,\ (\ :3/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.282909 + 0.689245i\)
\(L(\frac12)\) \(\approx\) \(0.282909 + 0.689245i\)
\(L(\frac{5}{2})\) 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 \( 1 + (13 + 45.0i)T \)
good3 \( 1 + (3.5 + 6.06i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 13.8iT - 125T^{2} \)
7 \( 1 + (19.5 + 11.2i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-19.5 + 11.2i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (13.5 - 23.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-76.5 - 44.1i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-28.5 - 49.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-34.5 - 59.7i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 72.7iT - 2.97e4T^{2} \)
37 \( 1 + (34.5 - 19.9i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (340.5 - 196. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (42.5 - 73.6i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 342. iT - 1.03e5T^{2} \)
53 \( 1 - 426T + 1.48e5T^{2} \)
59 \( 1 + (-16.5 - 9.52i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-8.5 + 14.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (142.5 - 82.2i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (505.5 + 291. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 1.00e3iT - 3.89e5T^{2} \)
79 \( 1 - 1.24e3T + 4.93e5T^{2} \)
83 \( 1 + 426. iT - 5.71e5T^{2} \)
89 \( 1 + (-265.5 + 153. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.06e3 - 617. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.76654393847799452367394449204, −10.35019045353818119101105120059, −9.329630798189467590957470575433, −8.145992344977311006261272464967, −7.12896518873452123874567985826, −6.15477055127752976397679991104, −5.16435217694280872395631433985, −3.51036244496939364932366121321, −1.32838220374691185642633726115, −0.37364276722848911651131215161, 2.69535879600867592654045281477, 3.87070445816458425015816254535, 5.15714871467645698005531691588, 6.41280930403358266020044442564, 7.07521820101085193053614385941, 9.071539953325130985305488354753, 9.732319424852190375846896016367, 10.53320130623585509782855630080, 11.46032346181254499125872958448, 12.15822848141272836543812984724

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