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Properties

Label 2-208-16.13-c1-0-18
Degree $2$
Conductor $208$
Sign $-0.141 + 0.989i$
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  + (0.0884 + 1.41i)2-s + (−1.08 − 1.08i)3-s + (−1.98 + 0.249i)4-s + (−2.20 + 2.20i)5-s + (1.43 − 1.62i)6-s − 4.01i·7-s + (−0.527 − 2.77i)8-s − 0.655i·9-s + (−3.30 − 2.91i)10-s + (−0.241 + 0.241i)11-s + (2.41 + 1.87i)12-s + (−0.707 − 0.707i)13-s + (5.67 − 0.355i)14-s + 4.76·15-s + (3.87 − 0.990i)16-s − 6.65·17-s + ⋯
L(s)  = 1  + (0.0625 + 0.998i)2-s + (−0.625 − 0.625i)3-s + (−0.992 + 0.124i)4-s + (−0.984 + 0.984i)5-s + (0.584 − 0.662i)6-s − 1.51i·7-s + (−0.186 − 0.982i)8-s − 0.218i·9-s + (−1.04 − 0.921i)10-s + (−0.0728 + 0.0728i)11-s + (0.698 + 0.542i)12-s + (−0.196 − 0.196i)13-s + (1.51 − 0.0949i)14-s + 1.23·15-s + (0.968 − 0.247i)16-s − 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.141 + 0.989i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ -0.141 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.186903 - 0.215627i\)
\(L(\frac12)\) \(\approx\) \(0.186903 - 0.215627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0884 - 1.41i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.08 + 1.08i)T + 3iT^{2} \)
5 \( 1 + (2.20 - 2.20i)T - 5iT^{2} \)
7 \( 1 + 4.01iT - 7T^{2} \)
11 \( 1 + (0.241 - 0.241i)T - 11iT^{2} \)
17 \( 1 + 6.65T + 17T^{2} \)
19 \( 1 + (3.76 + 3.76i)T + 19iT^{2} \)
23 \( 1 - 3.56iT - 23T^{2} \)
29 \( 1 + (-1.76 - 1.76i)T + 29iT^{2} \)
31 \( 1 - 4.18T + 31T^{2} \)
37 \( 1 + (5.81 - 5.81i)T - 37iT^{2} \)
41 \( 1 + 8.87iT - 41T^{2} \)
43 \( 1 + (-6.02 + 6.02i)T - 43iT^{2} \)
47 \( 1 + 2.34T + 47T^{2} \)
53 \( 1 + (1.04 - 1.04i)T - 53iT^{2} \)
59 \( 1 + (0.883 - 0.883i)T - 59iT^{2} \)
61 \( 1 + (-6.89 - 6.89i)T + 61iT^{2} \)
67 \( 1 + (9.50 + 9.50i)T + 67iT^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 + 7.82iT - 73T^{2} \)
79 \( 1 - 3.49T + 79T^{2} \)
83 \( 1 + (2.11 + 2.11i)T + 83iT^{2} \)
89 \( 1 - 8.48iT - 89T^{2} \)
97 \( 1 + 8.13T + 97T^{2} \)
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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

−12.20957776148299172533625737830, −11.10185323255456792496657299256, −10.39321089014059195295823371183, −8.848415033453089388921675883743, −7.51420960655281140478217504565, −7.01685697757911936303308183089, −6.40464600398332019852523368887, −4.62743982712833850999778861765, −3.60981930407405317911184911280, −0.25309293081823001122388136274, 2.33806728711702501355011974819, 4.24824904441663892092088119912, 4.83036920374114355529875187631, 5.98172970939945025669949429400, 8.292810426902282592846466494868, 8.709136327810653867987984826209, 9.837392970675135589887552971703, 11.02330246057950110544378515639, 11.64299554795945744602743085105, 12.41838316737981925920249243993

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