↑

Properties

Label 2-2912-8.5-c1-0-43
Degree $2$
Conductor $2912$
Sign $0.943 + 0.332i$
Analytic cond. $23.2524$
Root an. cond. $4.82207$
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  + 2.79i·3-s − 3.17i·5-s − 7-s − 4.83·9-s + 1.99i·11-s + i·13-s + 8.89·15-s + 7.81·17-s − 3.34i·19-s − 2.79i·21-s − 7.14·23-s − 5.10·25-s − 5.14i·27-s − 7.25i·29-s + 0.189·31-s + ⋯
L(s)  = 1  + 1.61i·3-s − 1.42i·5-s − 0.377·7-s − 1.61·9-s + 0.602i·11-s + 0.277i·13-s + 2.29·15-s + 1.89·17-s − 0.766i·19-s − 0.610i·21-s − 1.48·23-s − 1.02·25-s − 0.989i·27-s − 1.34i·29-s + 0.0340·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.943 + 0.332i$
Analytic conductor: \(23.2524\)
Root analytic conductor: \(4.82207\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2912,\ (\ :1/2),\ 0.943 + 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.405148039\)
\(L(\frac12)\) \(\approx\) \(1.405148039\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
13 \( 1 - iT \)
good3 \( 1 - 2.79iT - 3T^{2} \)
5 \( 1 + 3.17iT - 5T^{2} \)
11 \( 1 - 1.99iT - 11T^{2} \)
17 \( 1 - 7.81T + 17T^{2} \)
19 \( 1 + 3.34iT - 19T^{2} \)
23 \( 1 + 7.14T + 23T^{2} \)
29 \( 1 + 7.25iT - 29T^{2} \)
31 \( 1 - 0.189T + 31T^{2} \)
37 \( 1 + 8.57iT - 37T^{2} \)
41 \( 1 + 4.87T + 41T^{2} \)
43 \( 1 + 7.67iT - 43T^{2} \)
47 \( 1 + 2.25T + 47T^{2} \)
53 \( 1 - 7.38iT - 53T^{2} \)
59 \( 1 + 12.8iT - 59T^{2} \)
61 \( 1 - 12.6iT - 61T^{2} \)
67 \( 1 + 3.75iT - 67T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 - 5.64T + 79T^{2} \)
83 \( 1 + 10.6iT - 83T^{2} \)
89 \( 1 - 0.376T + 89T^{2} \)
97 \( 1 + 3.35T + 97T^{2} \)
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

−8.917016790770070364823789219519, −8.219409304767566518682380710582, −7.42759874521520306987898386321, −6.05068006178123661708293325470, −5.41123929208765754731643104882, −4.77213806634131827891909132748, −4.07700190476107138873203445323, −3.45446475607022449876186949257, −2.07305579736156247907775574374, −0.50834994216504563802127793184, 1.07164692751590625601421948987, 2.09740033219750236279730706675, 3.17421284866820870098911588968, 3.47279995544011647482348203922, 5.33185966146316360322952181733, 6.12296945746848297615113098483, 6.52690684045948292519486512213, 7.25359795707368849954042857412, 8.000478690613024368301293155656, 8.267582547724249082458742255773

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