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Properties

Label 2-40e2-8.5-c1-0-15
Degree $2$
Conductor $1600$
Sign $0.707 - 0.707i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  + 4.47·7-s + 3·9-s + 2i·11-s + 4.47i·13-s + 6i·19-s − 4.47·23-s − 4.47i·37-s + 2·41-s − 13.4·47-s + 13.0·49-s − 13.4i·53-s + 14i·59-s + 13.4·63-s + 8.94i·77-s + 9·81-s + ⋯
L(s)  = 1  + 1.69·7-s + 9-s + 0.603i·11-s + 1.24i·13-s + 1.37i·19-s − 0.932·23-s − 0.735i·37-s + 0.312·41-s − 1.95·47-s + 1.85·49-s − 1.84i·53-s + 1.82i·59-s + 1.69·63-s + 1.01i·77-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.183272904\)
\(L(\frac12)\) \(\approx\) \(2.183272904\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 3T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 4.47iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 4.47iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 13.4T + 47T^{2} \)
53 \( 1 + 13.4iT - 53T^{2} \)
59 \( 1 - 14iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.614202184221802678858818072598, −8.588929795657421364213050700276, −7.87591186707795253123565238636, −7.27134124376512838311962815932, −6.33569840776684366693071063117, −5.22851186037834574817226728988, −4.44073529687710282253373585693, −3.88357754307694610488811170521, −1.98277825287775790997044516532, −1.57784132869182798225866603749, 0.930536557049461890747862097621, 2.03559908497542300023086080287, 3.28333376198976661924085206796, 4.51258153787500963958187834931, 4.99470588143299034076198925000, 5.98144040077376839836244720847, 7.04069149780466421972864289643, 7.929688210517317700401464986393, 8.242841061494997062040123612907, 9.283380290711317816797606539210

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