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Properties

Label 2-208-16.13-c1-0-20
Degree $2$
Conductor $208$
Sign $0.408 + 0.912i$
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  + (1.38 − 0.285i)2-s + (−1.13 − 1.13i)3-s + (1.83 − 0.791i)4-s + (0.631 − 0.631i)5-s + (−1.90 − 1.25i)6-s + 0.0885i·7-s + (2.31 − 1.62i)8-s − 0.401i·9-s + (0.694 − 1.05i)10-s + (−2.57 + 2.57i)11-s + (−2.99 − 1.19i)12-s + (−0.707 − 0.707i)13-s + (0.0253 + 0.122i)14-s − 1.44·15-s + (2.74 − 2.90i)16-s + 5.17·17-s + ⋯
L(s)  = 1  + (0.979 − 0.202i)2-s + (−0.658 − 0.658i)3-s + (0.918 − 0.395i)4-s + (0.282 − 0.282i)5-s + (−0.777 − 0.511i)6-s + 0.0334i·7-s + (0.819 − 0.573i)8-s − 0.133i·9-s + (0.219 − 0.333i)10-s + (−0.776 + 0.776i)11-s + (−0.864 − 0.343i)12-s + (−0.196 − 0.196i)13-s + (0.00676 + 0.0327i)14-s − 0.371·15-s + (0.686 − 0.727i)16-s + 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.408 + 0.912i$
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: \((2,\ 208,\ (\ :1/2),\ 0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49772 - 0.970082i\)
\(L(\frac12)\) \(\approx\) \(1.49772 - 0.970082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.285i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.13 + 1.13i)T + 3iT^{2} \)
5 \( 1 + (-0.631 + 0.631i)T - 5iT^{2} \)
7 \( 1 - 0.0885iT - 7T^{2} \)
11 \( 1 + (2.57 - 2.57i)T - 11iT^{2} \)
17 \( 1 - 5.17T + 17T^{2} \)
19 \( 1 + (-0.0221 - 0.0221i)T + 19iT^{2} \)
23 \( 1 - 3.75iT - 23T^{2} \)
29 \( 1 + (-1.82 - 1.82i)T + 29iT^{2} \)
31 \( 1 + 1.92T + 31T^{2} \)
37 \( 1 + (6.60 - 6.60i)T - 37iT^{2} \)
41 \( 1 - 0.269iT - 41T^{2} \)
43 \( 1 + (1.21 - 1.21i)T - 43iT^{2} \)
47 \( 1 - 0.655T + 47T^{2} \)
53 \( 1 + (2.80 - 2.80i)T - 53iT^{2} \)
59 \( 1 + (-10.5 + 10.5i)T - 59iT^{2} \)
61 \( 1 + (7.11 + 7.11i)T + 61iT^{2} \)
67 \( 1 + (5.59 + 5.59i)T + 67iT^{2} \)
71 \( 1 - 3.73iT - 71T^{2} \)
73 \( 1 + 9.03iT - 73T^{2} \)
79 \( 1 + 0.755T + 79T^{2} \)
83 \( 1 + (7.09 + 7.09i)T + 83iT^{2} \)
89 \( 1 - 1.62iT - 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−12.36530995792825745959218830483, −11.63642083032614508584889996739, −10.49790883652204545269611639407, −9.569534853124307341223738809018, −7.73701755307022219500299287784, −6.90129630010752978357827425097, −5.70424703904923571869299565207, −5.02917775551823680159633903394, −3.31701629696266404771706285619, −1.56572484556364060953201780510, 2.63423615722940921984787000870, 4.08688678194071117675480809482, 5.30516650975929408865048302071, 5.91594360258451181404415098462, 7.25784036872324567993018279533, 8.390323418363113664412334059554, 10.19679725333630530192086559228, 10.63198343553132494687226941336, 11.66097684990094166610387581373, 12.52872754661743385547745849814

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