Properties

Label 2-42e2-28.27-c1-0-67
Degree $2$
Conductor $1764$
Sign $0.891 + 0.453i$
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  + (1.40 − 0.171i)2-s + (1.94 − 0.481i)4-s − 0.963i·5-s + (2.64 − 1.00i)8-s + (−0.165 − 1.35i)10-s + 5.48i·11-s − 3.75i·13-s + (3.53 − 1.87i)16-s − 0.686i·17-s + 4.88·19-s + (−0.464 − 1.87i)20-s + (0.941 + 7.69i)22-s + 1.24i·23-s + 4.07·25-s + (−0.643 − 5.26i)26-s + ⋯
L(s)  = 1  + (0.992 − 0.121i)2-s + (0.970 − 0.240i)4-s − 0.430i·5-s + (0.934 − 0.356i)8-s + (−0.0523 − 0.427i)10-s + 1.65i·11-s − 1.04i·13-s + (0.883 − 0.467i)16-s − 0.166i·17-s + 1.12·19-s + (−0.103 − 0.418i)20-s + (0.200 + 1.64i)22-s + 0.258i·23-s + 0.814·25-s + (−0.126 − 1.03i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.891 + 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.590202555\)
\(L(\frac12)\) \(\approx\) \(3.590202555\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.171i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.963iT - 5T^{2} \)
11 \( 1 - 5.48iT - 11T^{2} \)
13 \( 1 + 3.75iT - 13T^{2} \)
17 \( 1 + 0.686iT - 17T^{2} \)
19 \( 1 - 4.88T + 19T^{2} \)
23 \( 1 - 1.24iT - 23T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 - 4.82T + 31T^{2} \)
37 \( 1 + 2.73T + 37T^{2} \)
41 \( 1 + 9.42iT - 41T^{2} \)
43 \( 1 + 5.97iT - 43T^{2} \)
47 \( 1 + 3.61T + 47T^{2} \)
53 \( 1 - 4.09T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 10.4iT - 61T^{2} \)
67 \( 1 + 9.43iT - 67T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 - 6.66iT - 73T^{2} \)
79 \( 1 - 1.41iT - 79T^{2} \)
83 \( 1 + 0.543T + 83T^{2} \)
89 \( 1 + 0.554iT - 89T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−9.428664090933407676234577570650, −8.303850883890047497165948496151, −7.36181900335552167243953940391, −6.93695186496066414671209656132, −5.72231409763471653469372591397, −5.07660514584728772978539118463, −4.41561752031387405947484246859, −3.33553128952847161929662951067, −2.38358945119011199434539036595, −1.17803378896638368248984709221, 1.32402907241858470187908466907, 2.82145891216457744108759035923, 3.32774851587226308112256179170, 4.43062738213065909221589903790, 5.24856336398397313818723153279, 6.30424521134577451710782025082, 6.56869843195796362303669849184, 7.68408301038359337759567910459, 8.380468002996167887430444267477, 9.336126832414285716799290633978

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