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Properties

Label 2-208-16.13-c1-0-6
Degree $2$
Conductor $208$
Sign $-0.661 - 0.750i$
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  + (0.389 + 1.35i)2-s + (1.57 + 1.57i)3-s + (−1.69 + 1.05i)4-s + (−0.836 + 0.836i)5-s + (−1.52 + 2.75i)6-s − 0.226i·7-s + (−2.09 − 1.89i)8-s + 1.95i·9-s + (−1.46 − 0.811i)10-s + (1.47 − 1.47i)11-s + (−4.33 − 1.00i)12-s + (0.707 + 0.707i)13-s + (0.307 − 0.0880i)14-s − 2.63·15-s + (1.75 − 3.59i)16-s + 2.28·17-s + ⋯
L(s)  = 1  + (0.275 + 0.961i)2-s + (0.908 + 0.908i)3-s + (−0.848 + 0.529i)4-s + (−0.373 + 0.373i)5-s + (−0.623 + 1.12i)6-s − 0.0855i·7-s + (−0.742 − 0.670i)8-s + 0.651i·9-s + (−0.462 − 0.256i)10-s + (0.444 − 0.444i)11-s + (−1.25 − 0.290i)12-s + (0.196 + 0.196i)13-s + (0.0822 − 0.0235i)14-s − 0.679·15-s + (0.439 − 0.898i)16-s + 0.553·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.661 - 0.750i$
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: \((2,\ 208,\ (\ :1/2),\ -0.661 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.641995 + 1.42179i\)
\(L(\frac12)\) \(\approx\) \(0.641995 + 1.42179i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.389 - 1.35i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.57 - 1.57i)T + 3iT^{2} \)
5 \( 1 + (0.836 - 0.836i)T - 5iT^{2} \)
7 \( 1 + 0.226iT - 7T^{2} \)
11 \( 1 + (-1.47 + 1.47i)T - 11iT^{2} \)
17 \( 1 - 2.28T + 17T^{2} \)
19 \( 1 + (-1.90 - 1.90i)T + 19iT^{2} \)
23 \( 1 + 4.20iT - 23T^{2} \)
29 \( 1 + (-2.61 - 2.61i)T + 29iT^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 + (-1.39 + 1.39i)T - 37iT^{2} \)
41 \( 1 + 4.66iT - 41T^{2} \)
43 \( 1 + (2.77 - 2.77i)T - 43iT^{2} \)
47 \( 1 + 7.06T + 47T^{2} \)
53 \( 1 + (-6.38 + 6.38i)T - 53iT^{2} \)
59 \( 1 + (-5.30 + 5.30i)T - 59iT^{2} \)
61 \( 1 + (2.20 + 2.20i)T + 61iT^{2} \)
67 \( 1 + (2.95 + 2.95i)T + 67iT^{2} \)
71 \( 1 + 9.45iT - 71T^{2} \)
73 \( 1 + 1.62iT - 73T^{2} \)
79 \( 1 + 2.09T + 79T^{2} \)
83 \( 1 + (10.2 + 10.2i)T + 83iT^{2} \)
89 \( 1 - 10.1iT - 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−13.04137279430284913471666958600, −11.88275827462542454574456285729, −10.57261612091099404012444914075, −9.454173902697178523238497354581, −8.749631439934079073673138795530, −7.78986402357160964257669618690, −6.69117994103847602832265619313, −5.32916796032226697352036880318, −3.95777984701679865302189117286, −3.29624159535220378313514938763, 1.41308195301279698573636217318, 2.80829290911209313876192510116, 4.07415435451463160025971554005, 5.51772553944195786150321914280, 7.14963366590016739069200214240, 8.213060863064493113750321769015, 9.039481831051799839706670865848, 10.03617627354847938863774466972, 11.37090510335920351463165669756, 12.19135185351993385726291770027

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