↑

Properties

Label 2-40e2-20.7-c1-0-8
Degree $2$
Conductor $1600$
Sign $0.437 + 0.899i$
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  + (−2.22 − 2.22i)3-s + (−2 + 2i)7-s + 6.89i·9-s − 3.44i·11-s + (−4.44 + 4.44i)13-s + (−0.775 − 0.775i)17-s + 2.55·19-s + 8.89·21-s + (−0.449 − 0.449i)23-s + (8.67 − 8.67i)27-s − 2.89i·29-s + 2.89i·31-s + (−7.67 + 7.67i)33-s + (6 + 6i)37-s + 19.7·39-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)3-s + (−0.755 + 0.755i)7-s + 2.29i·9-s − 1.04i·11-s + (−1.23 + 1.23i)13-s + (−0.188 − 0.188i)17-s + 0.585·19-s + 1.94·21-s + (−0.0937 − 0.0937i)23-s + (1.66 − 1.66i)27-s − 0.538i·29-s + 0.520i·31-s + (−1.33 + 1.33i)33-s + (0.986 + 0.986i)37-s + 3.17·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6798213259\)
\(L(\frac12)\) \(\approx\) \(0.6798213259\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 2.22i)T + 3iT^{2} \)
7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 + 3.44iT - 11T^{2} \)
13 \( 1 + (4.44 - 4.44i)T - 13iT^{2} \)
17 \( 1 + (0.775 + 0.775i)T + 17iT^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 + (0.449 + 0.449i)T + 23iT^{2} \)
29 \( 1 + 2.89iT - 29T^{2} \)
31 \( 1 - 2.89iT - 31T^{2} \)
37 \( 1 + (-6 - 6i)T + 37iT^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + (2.89 + 2.89i)T + 43iT^{2} \)
47 \( 1 + (-7.34 + 7.34i)T - 47iT^{2} \)
53 \( 1 + (-6.44 + 6.44i)T - 53iT^{2} \)
59 \( 1 + 7.79T + 59T^{2} \)
61 \( 1 - 0.898T + 61T^{2} \)
67 \( 1 + (4.22 - 4.22i)T - 67iT^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (-5.67 + 5.67i)T - 73iT^{2} \)
79 \( 1 + 0.898T + 79T^{2} \)
83 \( 1 + (4.67 + 4.67i)T + 83iT^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 + (-4.89 - 4.89i)T + 97iT^{2} \)
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

−9.266950343539649070270945877580, −8.360252636972400012880584853287, −7.35254241838665144671351687620, −6.79301250209711840640584163462, −6.05397273512797337359265838590, −5.48803813828730591182458181258, −4.51738884655862349558140498924, −2.89901667049167228520355124873, −1.95591485598904206856418492912, −0.53714113357900642819312602688, 0.66394515891896171358083144374, 2.80395355190639347462902821278, 3.91689602141406857303457998534, 4.57369418392983996212544053728, 5.37602011770929950985856289247, 6.08987986884258970862303418479, 7.08786726759755931357954925202, 7.71474260984847764271635888698, 9.370745157865269852378816294016, 9.622214095009739007013205063083

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