Properties

Label 2-40e2-5.4-c1-0-28
Degree $2$
Conductor $1600$
Sign $-0.447 + 0.894i$
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 about

Normalization:  

Dirichlet series

L(s)  = 1  − 4i·7-s + 3·9-s − 4·11-s − 2i·13-s + 2i·17-s + 4·19-s − 4i·23-s − 2·29-s − 8·31-s − 6i·37-s − 6·41-s − 8i·43-s + 4i·47-s − 9·49-s + 6i·53-s + ⋯
L(s)  = 1  − 1.51i·7-s + 9-s − 1.20·11-s − 0.554i·13-s + 0.485i·17-s + 0.917·19-s − 0.834i·23-s − 0.371·29-s − 1.43·31-s − 0.986i·37-s − 0.937·41-s − 1.21i·43-s + 0.583i·47-s − 1.28·49-s + 0.824i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.276868154\)
\(L(\frac12)\) \(\approx\) \(1.276868154\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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

−9.270231862591860240656425884260, −8.140003332005757446198756522555, −7.39807223353356363562229469537, −7.10564226554235055524056984019, −5.83135606837318926877023171140, −4.92161818042559725828230006055, −4.06390688273939225570304188525, −3.24119016684682002831995332564, −1.79457675974558589641265003896, −0.48710251976651989983071855223, 1.63815028387202988186110531264, 2.61279107285671708396311829716, 3.62665358801877072391497326764, 5.03320658751984636300447285701, 5.33151221877542772152168056433, 6.43396732214031063076156644361, 7.36430741880210061266120750282, 8.017586346526220315568453851134, 9.017917827723684456430935223940, 9.572093029991495062249867714706

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