↑

Properties

Label 2-40e2-80.27-c1-0-15
Degree $2$
Conductor $1600$
Sign $0.982 + 0.186i$
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  − 2.55·3-s + (2.40 + 2.40i)7-s + 3.51·9-s + (2.67 − 2.67i)11-s − 2.40i·13-s + (0.0750 + 0.0750i)17-s + (−2.67 + 2.67i)19-s + (−6.13 − 6.13i)21-s + (2.12 − 2.12i)23-s − 1.30·27-s + (−3.95 − 3.95i)29-s + 1.65i·31-s + (−6.83 + 6.83i)33-s + 2.53i·37-s + 6.12i·39-s + ⋯
L(s)  = 1  − 1.47·3-s + (0.908 + 0.908i)7-s + 1.17·9-s + (0.807 − 0.807i)11-s − 0.666i·13-s + (0.0182 + 0.0182i)17-s + (−0.613 + 0.613i)19-s + (−1.33 − 1.33i)21-s + (0.442 − 0.442i)23-s − 0.250·27-s + (−0.734 − 0.734i)29-s + 0.297i·31-s + (−1.18 + 1.18i)33-s + 0.416i·37-s + 0.981i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.982 + 0.186i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.982 + 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.102908672\)
\(L(\frac12)\) \(\approx\) \(1.102908672\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.55T + 3T^{2} \)
7 \( 1 + (-2.40 - 2.40i)T + 7iT^{2} \)
11 \( 1 + (-2.67 + 2.67i)T - 11iT^{2} \)
13 \( 1 + 2.40iT - 13T^{2} \)
17 \( 1 + (-0.0750 - 0.0750i)T + 17iT^{2} \)
19 \( 1 + (2.67 - 2.67i)T - 19iT^{2} \)
23 \( 1 + (-2.12 + 2.12i)T - 23iT^{2} \)
29 \( 1 + (3.95 + 3.95i)T + 29iT^{2} \)
31 \( 1 - 1.65iT - 31T^{2} \)
37 \( 1 - 2.53iT - 37T^{2} \)
41 \( 1 - 1.70iT - 41T^{2} \)
43 \( 1 - 3.84iT - 43T^{2} \)
47 \( 1 + (-2.15 + 2.15i)T - 47iT^{2} \)
53 \( 1 - 1.29T + 53T^{2} \)
59 \( 1 + (5.29 + 5.29i)T + 59iT^{2} \)
61 \( 1 + (-10.2 + 10.2i)T - 61iT^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + 2.27T + 71T^{2} \)
73 \( 1 + (-9.99 - 9.99i)T + 73iT^{2} \)
79 \( 1 - 8.70T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (5.00 + 5.00i)T + 97iT^{2} \)
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

−9.422079906224486273789886726509, −8.504334566570369048978070625561, −7.894259646900848015016257477006, −6.60261748548607598832741788324, −6.07537242222916707003834048570, −5.35301828531968201185534156943, −4.70715951731147374962027097653, −3.51608493340211568872381746424, −2.02627565968541708661027708839, −0.74538312207570736370360606994, 0.901269725330391101393994113902, 1.94702025221634540437014673044, 3.88650487884306944330463040668, 4.55468536448825534035440672572, 5.22487162969953029632785664279, 6.22320460928395293348757777653, 7.04469370686934507613593693493, 7.42781685888512521070504302053, 8.748661152803250318264602463165, 9.521721730103310729783303938566

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