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Properties

Label 2-208-13.12-c1-0-3
Degree $2$
Conductor $208$
Sign $-0.155 + 0.987i$
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  − 1.56·3-s − 1.56i·5-s − 0.438i·7-s − 0.561·9-s − 5.12i·11-s + (0.561 − 3.56i)13-s + 2.43i·15-s − 3.56·17-s − 2i·19-s + 0.684i·21-s + 3.12·23-s + 2.56·25-s + 5.56·27-s − 5.12·29-s + 5.12i·31-s + ⋯
L(s)  = 1  − 0.901·3-s − 0.698i·5-s − 0.165i·7-s − 0.187·9-s − 1.54i·11-s + (0.155 − 0.987i)13-s + 0.629i·15-s − 0.863·17-s − 0.458i·19-s + 0.149i·21-s + 0.651·23-s + 0.512·25-s + 1.07·27-s − 0.951·29-s + 0.920i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.155 + 0.987i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ -0.155 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.468554 - 0.548219i\)
\(L(\frac12)\) \(\approx\) \(0.468554 - 0.548219i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.561 + 3.56i)T \)
good3 \( 1 + 1.56T + 3T^{2} \)
5 \( 1 + 1.56iT - 5T^{2} \)
7 \( 1 + 0.438iT - 7T^{2} \)
11 \( 1 + 5.12iT - 11T^{2} \)
17 \( 1 + 3.56T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 + 5.12T + 29T^{2} \)
31 \( 1 - 5.12iT - 31T^{2} \)
37 \( 1 - 9.56iT - 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 9.56T + 43T^{2} \)
47 \( 1 - 7.56iT - 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 - 9.12iT - 67T^{2} \)
71 \( 1 + 6.68iT - 71T^{2} \)
73 \( 1 + 9.36iT - 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 8.24iT - 83T^{2} \)
89 \( 1 + 3.12iT - 89T^{2} \)
97 \( 1 - 6.24iT - 97T^{2} \)
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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

−12.01213078717127680391296639029, −11.11138788263104641100889346134, −10.54343700672338964060102433609, −8.945106442960110903944438951950, −8.342872319307089119756477996582, −6.77588611208347028386675194576, −5.69588013458124994887251510723, −4.92739199494164754532049910346, −3.21363377967580390312145810641, −0.69192462608921522235827333748, 2.24747092535608378802647047353, 4.15082296394663614773849374791, 5.36112109606811661327052679404, 6.58775575131870785047016559507, 7.22985179894940092914701658185, 8.820820666356372044886541583390, 9.880899507032189909314234855502, 10.90602926067203601992047800609, 11.58329899034603382537160198332, 12.43783693583942543034238056687

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