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Properties

Label 2-208-16.13-c1-0-0
Degree $2$
Conductor $208$
Sign $0.145 - 0.989i$
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  + (−0.0872 − 1.41i)2-s + (−0.316 − 0.316i)3-s + (−1.98 + 0.246i)4-s + (−2.69 + 2.69i)5-s + (−0.419 + 0.474i)6-s + 1.94i·7-s + (0.521 + 2.78i)8-s − 2.79i·9-s + (4.04 + 3.56i)10-s + (−3.78 + 3.78i)11-s + (0.706 + 0.550i)12-s + (0.707 + 0.707i)13-s + (2.74 − 0.169i)14-s + 1.70·15-s + (3.87 − 0.978i)16-s − 1.40·17-s + ⋯
L(s)  = 1  + (−0.0617 − 0.998i)2-s + (−0.182 − 0.182i)3-s + (−0.992 + 0.123i)4-s + (−1.20 + 1.20i)5-s + (−0.171 + 0.193i)6-s + 0.735i·7-s + (0.184 + 0.982i)8-s − 0.933i·9-s + (1.27 + 1.12i)10-s + (−1.14 + 1.14i)11-s + (0.204 + 0.158i)12-s + (0.196 + 0.196i)13-s + (0.734 − 0.0454i)14-s + 0.440·15-s + (0.969 − 0.244i)16-s − 0.341·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.145 - 0.989i$
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: \((2,\ 208,\ (\ :1/2),\ 0.145 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299642 + 0.258901i\)
\(L(\frac12)\) \(\approx\) \(0.299642 + 0.258901i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0872 + 1.41i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.316 + 0.316i)T + 3iT^{2} \)
5 \( 1 + (2.69 - 2.69i)T - 5iT^{2} \)
7 \( 1 - 1.94iT - 7T^{2} \)
11 \( 1 + (3.78 - 3.78i)T - 11iT^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 + (2.81 + 2.81i)T + 19iT^{2} \)
23 \( 1 - 1.87iT - 23T^{2} \)
29 \( 1 + (-4.67 - 4.67i)T + 29iT^{2} \)
31 \( 1 + 5.23T + 31T^{2} \)
37 \( 1 + (-0.980 + 0.980i)T - 37iT^{2} \)
41 \( 1 - 6.55iT - 41T^{2} \)
43 \( 1 + (5.25 - 5.25i)T - 43iT^{2} \)
47 \( 1 + 3.38T + 47T^{2} \)
53 \( 1 + (-4.86 + 4.86i)T - 53iT^{2} \)
59 \( 1 + (-1.57 + 1.57i)T - 59iT^{2} \)
61 \( 1 + (4.55 + 4.55i)T + 61iT^{2} \)
67 \( 1 + (-8.66 - 8.66i)T + 67iT^{2} \)
71 \( 1 - 7.61iT - 71T^{2} \)
73 \( 1 + 13.6iT - 73T^{2} \)
79 \( 1 - 9.62T + 79T^{2} \)
83 \( 1 + (6.65 + 6.65i)T + 83iT^{2} \)
89 \( 1 - 9.24iT - 89T^{2} \)
97 \( 1 - 13.9T + 97T^{2} \)
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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

−12.37216010077451241775187203529, −11.59662944130715562748000083627, −10.90421663372380253912299255069, −9.930296980339549750487402929465, −8.760561953337302025174051752147, −7.64807634258155703551046891125, −6.58683560245952824703129717051, −4.87554343231417299866635368974, −3.57224690111602703499806832497, −2.47137847998375086927101219169, 0.34992777929780315188528427786, 3.86544808703641295166586410285, 4.78297170650494701757475099064, 5.72350229132598271936172142521, 7.36596541356538671224156308613, 8.203269229931745516272399541322, 8.605030765557005428242492931470, 10.24848553354266184207236704997, 11.05240702358021749307725383680, 12.42940958970432362207391917041

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