↑

Properties

Label 2-40e2-16.5-c1-0-1
Degree $2$
Conductor $1600$
Sign $-0.743 - 0.668i$
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  + (0.183 − 0.183i)3-s − 3.84i·7-s + 2.93i·9-s + (−1.60 − 1.60i)11-s + (−1.80 + 1.80i)13-s − 4.93·17-s + (−4.77 + 4.77i)19-s + (−0.707 − 0.707i)21-s + 0.134i·23-s + (1.09 + 1.09i)27-s + (−2.17 + 2.17i)29-s − 2.26·31-s − 0.588·33-s + (4.35 + 4.35i)37-s + 0.664i·39-s + ⋯
L(s)  = 1  + (0.106 − 0.106i)3-s − 1.45i·7-s + 0.977i·9-s + (−0.482 − 0.482i)11-s + (−0.501 + 0.501i)13-s − 1.19·17-s + (−1.09 + 1.09i)19-s + (−0.154 − 0.154i)21-s + 0.0280i·23-s + (0.209 + 0.209i)27-s + (−0.403 + 0.403i)29-s − 0.406·31-s − 0.102·33-s + (0.715 + 0.715i)37-s + 0.106i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.743 - 0.668i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.743 - 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3283335769\)
\(L(\frac12)\) \(\approx\) \(0.3283335769\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.183 + 0.183i)T - 3iT^{2} \)
7 \( 1 + 3.84iT - 7T^{2} \)
11 \( 1 + (1.60 + 1.60i)T + 11iT^{2} \)
13 \( 1 + (1.80 - 1.80i)T - 13iT^{2} \)
17 \( 1 + 4.93T + 17T^{2} \)
19 \( 1 + (4.77 - 4.77i)T - 19iT^{2} \)
23 \( 1 - 0.134iT - 23T^{2} \)
29 \( 1 + (2.17 - 2.17i)T - 29iT^{2} \)
31 \( 1 + 2.26T + 31T^{2} \)
37 \( 1 + (-4.35 - 4.35i)T + 37iT^{2} \)
41 \( 1 - 3.34iT - 41T^{2} \)
43 \( 1 + (2.70 + 2.70i)T + 43iT^{2} \)
47 \( 1 - 7.03T + 47T^{2} \)
53 \( 1 + (-3.40 - 3.40i)T + 53iT^{2} \)
59 \( 1 + (0.107 + 0.107i)T + 59iT^{2} \)
61 \( 1 + (3.46 - 3.46i)T - 61iT^{2} \)
67 \( 1 + (1.91 - 1.91i)T - 67iT^{2} \)
71 \( 1 - 9.32iT - 71T^{2} \)
73 \( 1 + 9.82iT - 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + (8.80 - 8.80i)T - 83iT^{2} \)
89 \( 1 + 1.12iT - 89T^{2} \)
97 \( 1 - 6.10T + 97T^{2} \)
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

−9.869837081803860999673429745658, −8.779529100840453852274548178312, −8.053984698987634576924365505422, −7.34621346606471375959242075136, −6.67850866452850755829821166460, −5.62194444298422849591042611450, −4.52333913121760525570097550619, −4.01620100383355144759781545473, −2.66540829217486931205579619891, −1.58348159442552750560086833185, 0.11579060296151630229680379700, 2.18610935657863910907295555368, 2.74771824576045063294576504537, 4.08433830725458944554913124087, 4.97896617732416650124811900016, 5.86428751021655806812750259537, 6.59854865832630102234007805939, 7.49563669535499565430365659250, 8.606598965376743893239650744118, 9.018137897133631191830268319481

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