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Properties

Label 2-208-13.11-c2-0-5
Degree $2$
Conductor $208$
Sign $0.335 - 0.941i$
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  + (1.52 + 2.63i)3-s + (4.79 + 4.79i)5-s + (−1.13 − 4.25i)7-s + (−0.132 + 0.228i)9-s + (13.8 + 3.71i)11-s + (1.84 + 12.8i)13-s + (−5.33 + 19.9i)15-s + (−20.9 − 12.1i)17-s + (−25.4 + 6.82i)19-s + (9.47 − 9.47i)21-s + (5.44 − 3.14i)23-s + 20.9i·25-s + 26.5·27-s + (11.1 + 19.2i)29-s + (8.59 + 8.59i)31-s + ⋯
L(s)  = 1  + (0.507 + 0.878i)3-s + (0.958 + 0.958i)5-s + (−0.162 − 0.607i)7-s + (−0.0146 + 0.0254i)9-s + (1.26 + 0.337i)11-s + (0.142 + 0.989i)13-s + (−0.355 + 1.32i)15-s + (−1.23 − 0.713i)17-s + (−1.33 + 0.359i)19-s + (0.451 − 0.451i)21-s + (0.236 − 0.136i)23-s + 0.836i·25-s + 0.984·27-s + (0.383 + 0.663i)29-s + (0.277 + 0.277i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.335 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.335 - 0.941i$
Analytic conductor: \(5.66758\)
Root analytic conductor: \(2.38066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1),\ 0.335 - 0.941i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.71167 + 1.20712i\)
\(L(\frac12)\) \(\approx\) \(1.71167 + 1.20712i\)
\(L(2)\) 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 \( 1 + (-1.84 - 12.8i)T \)
good3 \( 1 + (-1.52 - 2.63i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-4.79 - 4.79i)T + 25iT^{2} \)
7 \( 1 + (1.13 + 4.25i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-13.8 - 3.71i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (20.9 + 12.1i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (25.4 - 6.82i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-5.44 + 3.14i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.1 - 19.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-8.59 - 8.59i)T + 961iT^{2} \)
37 \( 1 + (6.13 + 1.64i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-18.8 + 70.4i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (26.9 + 15.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (7.65 - 7.65i)T - 2.20e3iT^{2} \)
53 \( 1 + 33.7T + 2.80e3T^{2} \)
59 \( 1 + (9.77 + 36.4i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-11.5 + 19.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-27.8 + 103. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (2.20 - 0.591i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-38.1 + 38.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 19.1T + 6.24e3T^{2} \)
83 \( 1 + (34.7 + 34.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-3.47 - 0.930i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (24.3 - 6.51i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−12.33196816331648971861330755896, −11.01957831272819131221152122449, −10.33987485100663655391046306966, −9.402519155499406385415962523584, −8.840682861892847390158798963615, −6.79103116942043400402041724702, −6.56280269779369730201825144477, −4.56125202153345300775009161045, −3.64008980613407851500697250871, −2.09581799566810024740097092273, 1.32605417531658485840145451916, 2.49916781929308801381948754805, 4.46377202022435693811153039194, 5.91401594805100994211415932143, 6.65505891530967420676911875327, 8.310456889523768794950337499203, 8.735698976889345074808328679797, 9.763367537657614203205045361884, 11.07875661362700986766734539435, 12.35327845718235881652624508893

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