Properties

Label 2-40e2-16.13-c1-0-4
Degree $2$
Conductor $1600$
Sign $0.125 - 0.992i$
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  + (−1.09 − 1.09i)3-s + 0.973i·7-s − 0.616i·9-s + (−1.40 + 1.40i)11-s + (4.60 + 4.60i)13-s − 0.490·17-s + (−4.54 − 4.54i)19-s + (1.06 − 1.06i)21-s + 1.94i·23-s + (−3.94 + 3.94i)27-s + (−3.74 − 3.74i)29-s − 4.29·31-s + 3.07·33-s + (−4.55 + 4.55i)37-s − 10.0i·39-s + ⋯
L(s)  = 1  + (−0.630 − 0.630i)3-s + 0.368i·7-s − 0.205i·9-s + (−0.424 + 0.424i)11-s + (1.27 + 1.27i)13-s − 0.118·17-s + (−1.04 − 1.04i)19-s + (0.231 − 0.231i)21-s + 0.405i·23-s + (−0.759 + 0.759i)27-s + (−0.695 − 0.695i)29-s − 0.770·31-s + 0.535·33-s + (−0.748 + 0.748i)37-s − 1.60i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.125 - 0.992i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.125 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7673402674\)
\(L(\frac12)\) \(\approx\) \(0.7673402674\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 + 1.09i)T + 3iT^{2} \)
7 \( 1 - 0.973iT - 7T^{2} \)
11 \( 1 + (1.40 - 1.40i)T - 11iT^{2} \)
13 \( 1 + (-4.60 - 4.60i)T + 13iT^{2} \)
17 \( 1 + 0.490T + 17T^{2} \)
19 \( 1 + (4.54 + 4.54i)T + 19iT^{2} \)
23 \( 1 - 1.94iT - 23T^{2} \)
29 \( 1 + (3.74 + 3.74i)T + 29iT^{2} \)
31 \( 1 + 4.29T + 31T^{2} \)
37 \( 1 + (4.55 - 4.55i)T - 37iT^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (1.79 - 1.79i)T - 43iT^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + (5.61 - 5.61i)T - 53iT^{2} \)
59 \( 1 + (8.44 - 8.44i)T - 59iT^{2} \)
61 \( 1 + (-3.01 - 3.01i)T + 61iT^{2} \)
67 \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \)
71 \( 1 - 0.897iT - 71T^{2} \)
73 \( 1 + 9.71iT - 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + (-0.815 - 0.815i)T + 83iT^{2} \)
89 \( 1 + 1.12iT - 89T^{2} \)
97 \( 1 - 7.54T + 97T^{2} \)
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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

−9.292281330076065425018813417148, −8.993779902046795726798074523471, −7.915651957017262135413569712531, −6.97679995432444255935117332191, −6.40279625063978813807292266023, −5.73551049854608195206932365114, −4.63562948323640054277194719233, −3.73814222611888508516948089154, −2.34811318097855214822238896000, −1.30380371382018237548730561490, 0.33633347443958154633652556922, 1.98516415032105604030895172822, 3.48867885384069417755086624660, 4.06349399070066887873289973374, 5.39070200596293672975718695477, 5.63093125053271601845336860967, 6.67802798081114681135856317148, 7.80100981330333180775054560419, 8.364079848879757514532401700543, 9.251260255554267755026097027303

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