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Properties

Label 2-42e2-12.11-c1-0-71
Degree $2$
Conductor $1764$
Sign $0.468 + 0.883i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
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  + (1.41 + 0.0900i)2-s + (1.98 + 0.254i)4-s − 2.48i·5-s + (2.77 + 0.537i)8-s + (0.224 − 3.51i)10-s − 4.60·11-s + 5.22·13-s + (3.87 + 1.00i)16-s − 5.61i·17-s − 3.19i·19-s + (0.632 − 4.93i)20-s + (−6.50 − 0.414i)22-s − 0.718·23-s − 1.19·25-s + (7.37 + 0.470i)26-s + ⋯
L(s)  = 1  + (0.997 + 0.0636i)2-s + (0.991 + 0.127i)4-s − 1.11i·5-s + (0.981 + 0.189i)8-s + (0.0708 − 1.11i)10-s − 1.38·11-s + 1.44·13-s + (0.967 + 0.252i)16-s − 1.36i·17-s − 0.732i·19-s + (0.141 − 1.10i)20-s + (−1.38 − 0.0884i)22-s − 0.149·23-s − 0.238·25-s + (1.44 + 0.0921i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.296279646\)
\(L(\frac12)\) \(\approx\) \(3.296279646\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0900i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.48iT - 5T^{2} \)
11 \( 1 + 4.60T + 11T^{2} \)
13 \( 1 - 5.22T + 13T^{2} \)
17 \( 1 + 5.61iT - 17T^{2} \)
19 \( 1 + 3.19iT - 19T^{2} \)
23 \( 1 + 0.718T + 23T^{2} \)
29 \( 1 - 4.53iT - 29T^{2} \)
31 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 - 2.71T + 37T^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + 11.1iT - 43T^{2} \)
47 \( 1 + 5.41T + 47T^{2} \)
53 \( 1 - 2.06iT - 53T^{2} \)
59 \( 1 - 4.11T + 59T^{2} \)
61 \( 1 + 1.01T + 61T^{2} \)
67 \( 1 - 12.6iT - 67T^{2} \)
71 \( 1 - 7.31T + 71T^{2} \)
73 \( 1 - 9.63T + 73T^{2} \)
79 \( 1 + 8.83iT - 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 - 8.52iT - 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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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

−8.965714477727808759697018957970, −8.358410540447531683519785534947, −7.47923140439676945421688501900, −6.68612448832016756594538602867, −5.51713401727138039679253688120, −5.19525384378440029461691370520, −4.32989390700934638483238258349, −3.29257860084576801088772098264, −2.29310300794628704004090109905, −0.902743975397020308276624471876, 1.65230077352672996673022997976, 2.76945929639834835961957556019, 3.48235965288210430590975210560, 4.33422371096945863535608791432, 5.52399439954500471442833565443, 6.18331265486865744504877601390, 6.71325247093647322726950760380, 7.982141582243576063532837988083, 8.120475234352533798195241144010, 9.805331881134269001148342841773

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