↑

Properties

Label 2-40e2-80.3-c1-0-10
Degree $2$
Conductor $1600$
Sign $-0.119 - 0.992i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.96·3-s + (−1.60 + 1.60i)7-s + 0.851·9-s + (−0.754 − 0.754i)11-s + 5.94i·13-s + (−1.95 + 1.95i)17-s + (−0.780 − 0.780i)19-s + (−3.14 + 3.14i)21-s + (4.93 + 4.93i)23-s − 4.21·27-s + (−1.44 + 1.44i)29-s − 3.60i·31-s + (−1.48 − 1.48i)33-s + 10.2i·37-s + 11.6i·39-s + ⋯
L(s)  = 1  + 1.13·3-s + (−0.605 + 0.605i)7-s + 0.283·9-s + (−0.227 − 0.227i)11-s + 1.64i·13-s + (−0.474 + 0.474i)17-s + (−0.179 − 0.179i)19-s + (−0.686 + 0.686i)21-s + (1.02 + 1.02i)23-s − 0.811·27-s + (−0.268 + 0.268i)29-s − 0.648i·31-s + (−0.257 − 0.257i)33-s + 1.68i·37-s + 1.86i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.119 - 0.992i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (943, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.119 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.783817543\)
\(L(\frac12)\) \(\approx\) \(1.783817543\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.96T + 3T^{2} \)
7 \( 1 + (1.60 - 1.60i)T - 7iT^{2} \)
11 \( 1 + (0.754 + 0.754i)T + 11iT^{2} \)
13 \( 1 - 5.94iT - 13T^{2} \)
17 \( 1 + (1.95 - 1.95i)T - 17iT^{2} \)
19 \( 1 + (0.780 + 0.780i)T + 19iT^{2} \)
23 \( 1 + (-4.93 - 4.93i)T + 23iT^{2} \)
29 \( 1 + (1.44 - 1.44i)T - 29iT^{2} \)
31 \( 1 + 3.60iT - 31T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 - 6.93iT - 41T^{2} \)
43 \( 1 + 9.91iT - 43T^{2} \)
47 \( 1 + (-0.104 - 0.104i)T + 47iT^{2} \)
53 \( 1 - 4.03T + 53T^{2} \)
59 \( 1 + (-3.46 + 3.46i)T - 59iT^{2} \)
61 \( 1 + (-0.680 - 0.680i)T + 61iT^{2} \)
67 \( 1 - 9.04iT - 67T^{2} \)
71 \( 1 - 3.64T + 71T^{2} \)
73 \( 1 + (-2.94 + 2.94i)T - 73iT^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 4.23T + 83T^{2} \)
89 \( 1 - 0.0426T + 89T^{2} \)
97 \( 1 + (-1.91 + 1.91i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.333273245534729043622044398964, −8.906661460212604262178286933010, −8.279520293370815066662035053120, −7.24306295253365215031827630919, −6.53205940224633402781306588401, −5.58849381096171278296311114133, −4.43382459793018376826980924520, −3.49613312943973142475407136095, −2.67424586162040054492325456581, −1.74684382625089251199861283553, 0.56426023197238445531546334508, 2.33534326247275415005372606428, 3.07526775708202024220071700012, 3.84774596389166772466575890280, 4.98473628339140962442292691934, 5.96156491310928835753816144254, 7.06371278568676327608583024730, 7.62839673559169062787424785552, 8.476814126668846806501090626333, 9.061176576999422791960149588456

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