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Properties

Label 2-40e2-80.69-c1-0-20
Degree $2$
Conductor $1600$
Sign $0.594 + 0.804i$
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  + (0.488 + 0.488i)3-s − 4.71·7-s − 2.52i·9-s + (3.91 + 3.91i)11-s + (−0.0878 − 0.0878i)13-s − 4.67i·17-s + (1.81 − 1.81i)19-s + (−2.30 − 2.30i)21-s − 1.63·23-s + (2.69 − 2.69i)27-s + (−3.26 + 3.26i)29-s + 2.12·31-s + 3.82i·33-s + (3.97 − 3.97i)37-s − 0.0858i·39-s + ⋯
L(s)  = 1  + (0.282 + 0.282i)3-s − 1.78·7-s − 0.840i·9-s + (1.17 + 1.17i)11-s + (−0.0243 − 0.0243i)13-s − 1.13i·17-s + (0.415 − 0.415i)19-s + (−0.502 − 0.502i)21-s − 0.339·23-s + (0.519 − 0.519i)27-s + (−0.606 + 0.606i)29-s + 0.382·31-s + 0.665i·33-s + (0.653 − 0.653i)37-s − 0.0137i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.594 + 0.804i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.594 + 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.357772261\)
\(L(\frac12)\) \(\approx\) \(1.357772261\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.488 - 0.488i)T + 3iT^{2} \)
7 \( 1 + 4.71T + 7T^{2} \)
11 \( 1 + (-3.91 - 3.91i)T + 11iT^{2} \)
13 \( 1 + (0.0878 + 0.0878i)T + 13iT^{2} \)
17 \( 1 + 4.67iT - 17T^{2} \)
19 \( 1 + (-1.81 + 1.81i)T - 19iT^{2} \)
23 \( 1 + 1.63T + 23T^{2} \)
29 \( 1 + (3.26 - 3.26i)T - 29iT^{2} \)
31 \( 1 - 2.12T + 31T^{2} \)
37 \( 1 + (-3.97 + 3.97i)T - 37iT^{2} \)
41 \( 1 + 8.25iT - 41T^{2} \)
43 \( 1 + (-2.27 + 2.27i)T - 43iT^{2} \)
47 \( 1 + 4.06iT - 47T^{2} \)
53 \( 1 + (-5.03 + 5.03i)T - 53iT^{2} \)
59 \( 1 + (5.16 + 5.16i)T + 59iT^{2} \)
61 \( 1 + (-7.12 + 7.12i)T - 61iT^{2} \)
67 \( 1 + (7.49 + 7.49i)T + 67iT^{2} \)
71 \( 1 + 4.54iT - 71T^{2} \)
73 \( 1 - 8.30T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + (-1.16 - 1.16i)T + 83iT^{2} \)
89 \( 1 + 3.24iT - 89T^{2} \)
97 \( 1 - 13.9iT - 97T^{2} \)
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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

−9.461264393612599397073588738297, −8.940689834973644084372778152021, −7.45523130370712915303177344544, −6.79250017731106240959334674714, −6.31995848728357088430580797580, −5.12602010536482251791613243598, −3.92877871876725428828165962600, −3.44747510774273819988696551347, −2.33445762673132776451860395137, −0.57041123053864873518910872855, 1.18776013242185260190915097868, 2.65265568736742455409684922352, 3.47079538777307163282359323344, 4.26366175228022466339278895014, 5.90165860322780381788152821844, 6.12072616558178745350917673226, 7.06387666628999102618541251206, 8.019106642858102778995174471957, 8.722683415880119426432127747015, 9.535188694189700429620423824498

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