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Properties

Label 2-208-13.4-c1-0-4
Degree $2$
Conductor $208$
Sign $0.604 + 0.796i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.13 − 1.97i)3-s + 1.12i·5-s + (2.26 − 1.30i)7-s + (−1.09 − 1.89i)9-s + (−2.26 − 1.30i)11-s + (3.30 + 1.43i)13-s + (2.21 + 1.27i)15-s + (−2.77 − 4.81i)17-s + (−6.88 + 3.97i)19-s − 5.94i·21-s + (1.13 − 1.97i)23-s + 3.74·25-s + 1.85·27-s + (−3.89 + 6.75i)29-s + 8.26i·31-s + ⋯
L(s)  = 1  + (0.657 − 1.13i)3-s + 0.501i·5-s + (0.854 − 0.493i)7-s + (−0.364 − 0.631i)9-s + (−0.681 − 0.393i)11-s + (0.916 + 0.399i)13-s + (0.571 + 0.329i)15-s + (−0.673 − 1.16i)17-s + (−1.57 + 0.911i)19-s − 1.29i·21-s + (0.237 − 0.411i)23-s + 0.748·25-s + 0.356·27-s + (−0.724 + 1.25i)29-s + 1.48i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.604 + 0.796i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ 0.604 + 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35459 - 0.672270i\)
\(L(\frac12)\) \(\approx\) \(1.35459 - 0.672270i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.30 - 1.43i)T \)
good3 \( 1 + (-1.13 + 1.97i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 1.12iT - 5T^{2} \)
7 \( 1 + (-2.26 + 1.30i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.26 + 1.30i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.77 + 4.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.88 - 3.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.13 + 1.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.89 - 6.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.26iT - 31T^{2} \)
37 \( 1 + (-3.11 - 1.80i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.96 - 2.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.01 + 3.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.66iT - 47T^{2} \)
53 \( 1 + 9.54T + 53T^{2} \)
59 \( 1 + (3.31 - 1.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.84 + 6.65i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.52 - 0.881i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.880 - 0.508i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.323iT - 73T^{2} \)
79 \( 1 - 2.66T + 79T^{2} \)
83 \( 1 + 2.77iT - 83T^{2} \)
89 \( 1 + (9.99 + 5.77i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.5 + 7.82i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.52239168423078287463373310884, −11.08677907096824652848572694078, −10.67353840630671506235451388080, −8.878179513958120133596460909018, −8.178370435045032878055624055630, −7.21906181881715261039024378397, −6.39534464791957759049832995561, −4.69609159374443154259389795463, −3.01177033211105591350029734264, −1.62617376035847109654437406016, 2.30613937214524220696120401261, 4.01701831214663217520906211162, 4.79651180457003162077065990064, 6.10184876469034710110198126682, 7.981645021382318139199207377810, 8.637628710680529487264477016630, 9.425698354508830926956291477361, 10.65071151441052713082674113652, 11.20907870135268447472147883418, 12.76116557336258768946474461907

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