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Properties

Label 2-208-13.3-c1-0-4
Degree $2$
Conductor $208$
Sign $0.128 + 0.991i$
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  + (−0.780 − 1.35i)3-s + 0.561·5-s + (0.780 − 1.35i)7-s + (0.280 − 0.486i)9-s + (−0.780 − 1.35i)11-s + (−2.84 − 2.21i)13-s + (−0.438 − 0.759i)15-s + (2.5 − 4.33i)17-s + (0.780 − 1.35i)19-s − 2.43·21-s + (4.34 + 7.52i)23-s − 4.68·25-s − 5.56·27-s + (2.5 + 4.33i)29-s + 8·31-s + ⋯
L(s)  = 1  + (−0.450 − 0.780i)3-s + 0.251·5-s + (0.295 − 0.511i)7-s + (0.0935 − 0.162i)9-s + (−0.235 − 0.407i)11-s + (−0.788 − 0.615i)13-s + (−0.113 − 0.196i)15-s + (0.606 − 1.05i)17-s + (0.179 − 0.310i)19-s − 0.532·21-s + (0.905 + 1.56i)23-s − 0.936·25-s − 1.07·27-s + (0.464 + 0.804i)29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.128 + 0.991i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ 0.128 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.803102 - 0.705590i\)
\(L(\frac12)\) \(\approx\) \(0.803102 - 0.705590i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.84 + 2.21i)T \)
good3 \( 1 + (0.780 + 1.35i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 0.561T + 5T^{2} \)
7 \( 1 + (-0.780 + 1.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.780 + 1.35i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.780 + 1.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.34 - 7.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.62 - 6.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.21 - 2.11i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 8.56T + 53T^{2} \)
59 \( 1 + (-0.780 + 1.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.62 - 8.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.78 + 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.34 - 4.05i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 14.2T + 83T^{2} \)
89 \( 1 + (1.34 + 2.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.90 + 15.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.06239437630389849899639796528, −11.45741161185124669555293981377, −10.22045217581594147603991139179, −9.333600285491245803022014415242, −7.77525809220908633429278080640, −7.19778579974932822272382337339, −5.94823987104341799490845436024, −4.88101596639276508057014703057, −3.05398872855925400530140787959, −1.06963595628060547927690318413, 2.26865970067043317401128083191, 4.21314387086822492373958705255, 5.11400429893814385374725378895, 6.21574665652626670462219444635, 7.62600854714932084000845742647, 8.762958235711035295604279882441, 9.988675822597781442327212172118, 10.41313488751471618930380456142, 11.67171134290994862471002697089, 12.39425756204273551798662772128

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