↑

Properties

Label 2-40e2-16.13-c1-0-28
Degree $2$
Conductor $1600$
Sign $0.382 + 0.923i$
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  + (1.15 + 1.15i)3-s − 4.31i·7-s − 0.316i·9-s + (0.158 − 0.158i)11-s + (2.31 + 2.31i)13-s − 5.31·17-s + (−3.15 − 3.15i)19-s + (5 − 5i)21-s − 6.31i·23-s + (3.84 − 3.84i)27-s + (2 + 2i)29-s − 4.31·31-s + 0.366·33-s + (7.31 − 7.31i)37-s + 5.36i·39-s + ⋯
L(s)  = 1  + (0.668 + 0.668i)3-s − 1.63i·7-s − 0.105i·9-s + (0.0477 − 0.0477i)11-s + (0.642 + 0.642i)13-s − 1.28·17-s + (−0.724 − 0.724i)19-s + (1.09 − 1.09i)21-s − 1.31i·23-s + (0.739 − 0.739i)27-s + (0.371 + 0.371i)29-s − 0.775·31-s + 0.0638·33-s + (1.20 − 1.20i)37-s + 0.859i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.809248479\)
\(L(\frac12)\) \(\approx\) \(1.809248479\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 - 1.15i)T + 3iT^{2} \)
7 \( 1 + 4.31iT - 7T^{2} \)
11 \( 1 + (-0.158 + 0.158i)T - 11iT^{2} \)
13 \( 1 + (-2.31 - 2.31i)T + 13iT^{2} \)
17 \( 1 + 5.31T + 17T^{2} \)
19 \( 1 + (3.15 + 3.15i)T + 19iT^{2} \)
23 \( 1 + 6.31iT - 23T^{2} \)
29 \( 1 + (-2 - 2i)T + 29iT^{2} \)
31 \( 1 + 4.31T + 31T^{2} \)
37 \( 1 + (-7.31 + 7.31i)T - 37iT^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 + (5.63 - 5.63i)T - 43iT^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (-3.31 + 3.31i)T - 53iT^{2} \)
59 \( 1 + (5.31 - 5.31i)T - 59iT^{2} \)
61 \( 1 + (3.63 + 3.63i)T + 61iT^{2} \)
67 \( 1 + (5.84 + 5.84i)T + 67iT^{2} \)
71 \( 1 - 4.63iT - 71T^{2} \)
73 \( 1 + 13.3iT - 73T^{2} \)
79 \( 1 - 2.31T + 79T^{2} \)
83 \( 1 + (-3.84 - 3.84i)T + 83iT^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 - 6.63T + 97T^{2} \)
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

−9.040348563326951090120141437847, −8.799678684226079884228985180967, −7.69210409137404086386818568254, −6.80207046743745894820754360355, −6.29348190100805176253806152165, −4.56524291205268697569200895763, −4.27187274539724646253298654853, −3.45220310801943847499178651773, −2.22502577993279540396363817023, −0.62888714213558903279079969577, 1.64993656369309558021626856825, 2.41925888802632049721144860717, 3.28622693289934310391475457305, 4.60303298702409387919673704310, 5.67712232131631853341167579835, 6.22537114419871080106546209066, 7.29549711125086325765133763557, 8.174776515762829032561459837307, 8.627187060413234491113883551229, 9.250685323339087417539975474037

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