Properties

Label 2-40e2-5.4-c1-0-20
Degree $2$
Conductor $1600$
Sign $0.894 + 0.447i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − 2i·7-s + 2·9-s + 5·11-s − 5i·17-s − 5·19-s + 2·21-s − 6i·23-s + 5i·27-s + 4·29-s − 10·31-s + 5i·33-s − 10i·37-s + 5·41-s + 4i·43-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.755i·7-s + 0.666·9-s + 1.50·11-s − 1.21i·17-s − 1.14·19-s + 0.436·21-s − 1.25i·23-s + 0.962i·27-s + 0.742·29-s − 1.79·31-s + 0.870i·33-s − 1.64i·37-s + 0.780·41-s + 0.609i·43-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
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: \((2,\ 1600,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.864861475\)
\(L(\frac12)\) \(\approx\) \(1.864861475\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 5iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + iT - 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.165726533024305486658527405683, −8.968561195121455879911428114774, −7.58813209821588575866016272332, −6.95860533905878469011456021695, −6.25332556535095987565649285236, −4.98587120133807226879397449288, −4.17500832262042093160340389994, −3.71379414996926114056542173444, −2.20180909365265860431114440790, −0.812837959421356350827903639239, 1.35225898026343701835050404094, 2.09809972258832318742786203106, 3.59696575161468210870095631863, 4.30376623456525155701418241215, 5.55457233241086900779968281508, 6.38920341941992784175220056308, 6.88933176995308127332384693969, 7.923179773704908364355655796203, 8.670977469165622527725920724351, 9.358489985756048941518706710249

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