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Properties

Label 2-208-13.4-c1-0-0
Degree $2$
Conductor $208$
Sign $-0.950 - 0.311i$
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  + (−1.63 + 2.83i)3-s + 2.61i·5-s + (0.971 − 0.560i)7-s + (−3.87 − 6.70i)9-s + (−0.971 − 0.560i)11-s + (−3.53 + 0.694i)13-s + (−7.40 − 4.27i)15-s + (2.77 + 4.81i)17-s + (1.45 − 0.838i)19-s + 3.67i·21-s + (−1.63 + 2.83i)23-s − 1.81·25-s + 15.5·27-s + (0.167 − 0.289i)29-s + 0.129i·31-s + ⋯
L(s)  = 1  + (−0.946 + 1.63i)3-s + 1.16i·5-s + (0.367 − 0.212i)7-s + (−1.29 − 2.23i)9-s + (−0.292 − 0.169i)11-s + (−0.981 + 0.192i)13-s + (−1.91 − 1.10i)15-s + (0.673 + 1.16i)17-s + (0.333 − 0.192i)19-s + 0.802i·21-s + (−0.341 + 0.591i)23-s − 0.362·25-s + 2.99·27-s + (0.0310 − 0.0537i)29-s + 0.0232i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.950 - 0.311i$
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: \((2,\ 208,\ (\ :1/2),\ -0.950 - 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116891 + 0.731685i\)
\(L(\frac12)\) \(\approx\) \(0.116891 + 0.731685i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.53 - 0.694i)T \)
good3 \( 1 + (1.63 - 2.83i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.61iT - 5T^{2} \)
7 \( 1 + (-0.971 + 0.560i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.971 + 0.560i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.77 - 4.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.45 + 0.838i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.63 - 2.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.167 + 0.289i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.129iT - 31T^{2} \)
37 \( 1 + (3.92 + 2.26i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.96 - 2.86i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.24 - 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.7iT - 47T^{2} \)
53 \( 1 - 4.14T + 53T^{2} \)
59 \( 1 + (-0.549 + 0.317i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.35 + 4.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.4 - 7.18i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.45 + 4.30i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 9.94iT - 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + 5.75iT - 83T^{2} \)
89 \( 1 + (-12.4 - 7.17i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.80 + 4.50i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.47315810127910866813392526095, −11.44323798442149274556001056423, −10.79957724225889586673771535398, −10.15003734275831583569653079183, −9.345277967897448408681135464748, −7.74758622582246986155518290482, −6.38807865037430810698870958406, −5.41581344579565868853675196266, −4.27365011385025025216680557649, −3.12855687377383276190783517412, 0.71788585314615945072148133478, 2.23659716465380457153040365094, 5.01364652154310029140444960555, 5.47719237504946267947082057117, 6.93777407895479820458092954369, 7.73955450511475976791890118721, 8.652802107598624169417400803463, 10.10230948797298557159929647414, 11.47642950066498149242798304252, 12.17773587531630144081129744689

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