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Properties

Label 2-42e2-28.27-c1-0-21
Degree $2$
Conductor $1764$
Sign $-0.823 - 0.567i$
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  + (0.0777 + 1.41i)2-s + (−1.98 + 0.219i)4-s − 0.438i·5-s + (−0.464 − 2.79i)8-s + (0.619 − 0.0341i)10-s + 2.11i·11-s + 3.84i·13-s + (3.90 − 0.872i)16-s − 5.64i·17-s + 2.97·19-s + (0.0963 + 0.872i)20-s + (−2.98 + 0.164i)22-s + 4.77i·23-s + 4.80·25-s + (−5.43 + 0.299i)26-s + ⋯
L(s)  = 1  + (0.0549 + 0.998i)2-s + (−0.993 + 0.109i)4-s − 0.196i·5-s + (−0.164 − 0.986i)8-s + (0.196 − 0.0107i)10-s + 0.637i·11-s + 1.06i·13-s + (0.975 − 0.218i)16-s − 1.36i·17-s + 0.682·19-s + (0.0215 + 0.195i)20-s + (−0.637 + 0.0350i)22-s + 0.994i·23-s + 0.961·25-s + (−1.06 + 0.0586i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.823 - 0.567i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.270011896\)
\(L(\frac12)\) \(\approx\) \(1.270011896\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0777 - 1.41i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.438iT - 5T^{2} \)
11 \( 1 - 2.11iT - 11T^{2} \)
13 \( 1 - 3.84iT - 13T^{2} \)
17 \( 1 + 5.64iT - 17T^{2} \)
19 \( 1 - 2.97T + 19T^{2} \)
23 \( 1 - 4.77iT - 23T^{2} \)
29 \( 1 + 7.02T + 29T^{2} \)
31 \( 1 - 7.42T + 31T^{2} \)
37 \( 1 + 5.28T + 37T^{2} \)
41 \( 1 - 6.81iT - 41T^{2} \)
43 \( 1 - 4.38iT - 43T^{2} \)
47 \( 1 + 1.68T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 8.11T + 59T^{2} \)
61 \( 1 - 6.18iT - 61T^{2} \)
67 \( 1 - 7.85iT - 67T^{2} \)
71 \( 1 - 1.16iT - 71T^{2} \)
73 \( 1 - 10.0iT - 73T^{2} \)
79 \( 1 - 15.5iT - 79T^{2} \)
83 \( 1 + 5.49T + 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 + 2.22iT - 97T^{2} \)
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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

−9.584323681429489200532843326245, −8.801103983294164339300125525296, −7.896098822544746890111663888793, −7.11408340608477506028441877986, −6.67070872870219406373962923313, −5.49391939004595321075380442094, −4.87602989593395492775705219357, −4.07397153120434129206504001316, −2.90034902020883849561455822670, −1.27008977054413809012530461629, 0.52398347610943382622592652224, 1.84365724635756628154704672414, 3.04551665220088514370227104142, 3.62270308483681287694916659572, 4.75638998497639675439155537045, 5.59192613524347634279021569837, 6.38897426866420210436096047039, 7.66739499694235469917738928934, 8.402469294171699232642030967661, 8.981641670722713788921856493634

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