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Properties

Label 2-208-13.9-c1-0-3
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)3-s − 3·5-s + (−2 − 3.46i)7-s + (−0.499 − 0.866i)9-s + (−3.5 − 0.866i)13-s + (3 − 5.19i)15-s + (−1.5 − 2.59i)17-s + (1 + 1.73i)19-s + 7.99·21-s + (−3 + 5.19i)23-s + 4·25-s − 4.00·27-s + (−4.5 + 7.79i)29-s − 2·31-s + (6 + 10.3i)35-s + ⋯
L(s)  = 1  + (−0.577 + 0.999i)3-s − 1.34·5-s + (−0.755 − 1.30i)7-s + (−0.166 − 0.288i)9-s + (−0.970 − 0.240i)13-s + (0.774 − 1.34i)15-s + (−0.363 − 0.630i)17-s + (0.229 + 0.397i)19-s + 1.74·21-s + (−0.625 + 1.08i)23-s + 0.800·25-s − 0.769·27-s + (−0.835 + 1.44i)29-s − 0.359·31-s + (1.01 + 1.75i)35-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: \(1\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ -0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.76723313813513363110487570734, −10.95181154227276611127910041209, −10.18302904325864607439820242012, −9.323751804597702879511898195565, −7.64552445454731386284228952085, −7.14079469191060000414280246190, −5.37807324050338824368919403796, −4.21980828304961815269465349177, −3.53353645421220538809057009693, 0, 2.47848947415462684216597667230, 4.16016509383550049891584683159, 5.74529536527733475310763539183, 6.69129028357017168615477371580, 7.63768284863862900034135019395, 8.643291408338096958006842452163, 9.814504439695192036163287981354, 11.35007561437006784350373566171, 12.01780262806743103088272019846

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