Properties

Label 2-84e2-21.20-c1-0-1
Degree $2$
Conductor $7056$
Sign $-0.716 - 0.698i$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
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  − 3.37·5-s − 0.828i·11-s − 3.37i·13-s − 1.39·17-s + 6.75i·19-s − 2i·23-s + 6.41·25-s − 4.82i·29-s − 6.75i·31-s − 2.58·37-s − 8.15·41-s + 12.4·43-s + 6.75·47-s − 7.07i·53-s + 2.79i·55-s + ⋯
L(s)  = 1  − 1.51·5-s − 0.249i·11-s − 0.937i·13-s − 0.339·17-s + 1.55i·19-s − 0.417i·23-s + 1.28·25-s − 0.896i·29-s − 1.21i·31-s − 0.425·37-s − 1.27·41-s + 1.90·43-s + 0.985·47-s − 0.971i·53-s + 0.377i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.716 - 0.698i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -0.716 - 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1765134530\)
\(L(\frac12)\) \(\approx\) \(0.1765134530\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3.37T + 5T^{2} \)
11 \( 1 + 0.828iT - 11T^{2} \)
13 \( 1 + 3.37iT - 13T^{2} \)
17 \( 1 + 1.39T + 17T^{2} \)
19 \( 1 - 6.75iT - 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 + 4.82iT - 29T^{2} \)
31 \( 1 + 6.75iT - 31T^{2} \)
37 \( 1 + 2.58T + 37T^{2} \)
41 \( 1 + 8.15T + 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 - 6.75T + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 - 6.75T + 59T^{2} \)
61 \( 1 - 8.15iT - 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 + 4.82iT - 71T^{2} \)
73 \( 1 - 1.39iT - 73T^{2} \)
79 \( 1 - 9.65T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 6.17T + 89T^{2} \)
97 \( 1 + 1.39iT - 97T^{2} \)
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

−8.133341375337232911795274007996, −7.65161473611829646923869581534, −6.99870509291527148129901174437, −6.01221287356759935072251979740, −5.48263038293373786777099821456, −4.36151740946110335605477083651, −3.94427879919725795316477955228, −3.22179582374106985687948057846, −2.26776952378465629430712287428, −0.882724409831705366553607951473, 0.05743720043484823956650769982, 1.29191832746272161862674408278, 2.51824631695324832158276186123, 3.35198298586698080503832491603, 4.13797751775257666169675402226, 4.65290829692977063682745093839, 5.39276507201394412616678059919, 6.57772722754232507757624799593, 7.12647919504995425818612047648, 7.46030339165219011867953745039

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