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Properties

Label 2-208-13.9-c1-0-1
Degree $2$
Conductor $208$
Sign $0.859 - 0.511i$
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  − 5-s + (2 + 3.46i)7-s + (1.5 + 2.59i)9-s + (2 − 3.46i)11-s + (3.5 + 0.866i)13-s + (−1.5 − 2.59i)17-s + (−2 + 3.46i)23-s − 4·25-s + (0.5 − 0.866i)29-s − 4·31-s + (−2 − 3.46i)35-s + (−1.5 + 2.59i)37-s + (4.5 − 7.79i)41-s + (−4 − 6.92i)43-s + (−1.5 − 2.59i)45-s + ⋯
L(s)  = 1  − 0.447·5-s + (0.755 + 1.30i)7-s + (0.5 + 0.866i)9-s + (0.603 − 1.04i)11-s + (0.970 + 0.240i)13-s + (−0.363 − 0.630i)17-s + (−0.417 + 0.722i)23-s − 0.800·25-s + (0.0928 − 0.160i)29-s − 0.718·31-s + (−0.338 − 0.585i)35-s + (−0.246 + 0.427i)37-s + (0.702 − 1.21i)41-s + (−0.609 − 1.05i)43-s + (−0.223 − 0.387i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21009 + 0.332575i\)
\(L(\frac12)\) \(\approx\) \(1.21009 + 0.332575i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4 + 6.92i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.25196933517873443161541911548, −11.46712237280786853388564830075, −10.85714630926024950776075283777, −9.278124938169969448763987916790, −8.512862190502260882779457285438, −7.61653422691277270803106023594, −6.12242102580676883025405456319, −5.12611426549667021393662749119, −3.71877137429415820204304026776, −1.95416902815962143060931653700, 1.37657063688606665036298754407, 3.85528860880685358985131670949, 4.40410805845979791476760715651, 6.29156876859747315138332365657, 7.26066693892898955863983143934, 8.163339103320999247101422951462, 9.425870767464324672988072167903, 10.45902203114330446642378420660, 11.27538744150582375657188587471, 12.31254166152761015895881843902

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