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Properties

Label 2-42e2-28.27-c1-0-6
Degree $2$
Conductor $1764$
Sign $-0.436 - 0.899i$
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  + (−0.207 − 1.39i)2-s + (−1.91 + 0.579i)4-s + 2.61i·5-s + (1.20 + 2.55i)8-s + (3.65 − 0.541i)10-s + 3.95i·11-s − 1.08i·13-s + (3.32 − 2.21i)16-s + 0.317i·17-s − 5.16·19-s + (−1.51 − 5.00i)20-s + (5.53 − 0.819i)22-s + 2.31i·23-s − 1.82·25-s + (−1.51 + 0.224i)26-s + ⋯
L(s)  = 1  + (−0.146 − 0.989i)2-s + (−0.957 + 0.289i)4-s + 1.16i·5-s + (0.426 + 0.904i)8-s + (1.15 − 0.171i)10-s + 1.19i·11-s − 0.300i·13-s + (0.832 − 0.554i)16-s + 0.0768i·17-s − 1.18·19-s + (−0.338 − 1.11i)20-s + (1.18 − 0.174i)22-s + 0.483i·23-s − 0.365·25-s + (−0.296 + 0.0439i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.436 - 0.899i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.436 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5719457050\)
\(L(\frac12)\) \(\approx\) \(0.5719457050\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.207 + 1.39i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2.61iT - 5T^{2} \)
11 \( 1 - 3.95iT - 11T^{2} \)
13 \( 1 + 1.08iT - 13T^{2} \)
17 \( 1 - 0.317iT - 17T^{2} \)
19 \( 1 + 5.16T + 19T^{2} \)
23 \( 1 - 2.31iT - 23T^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
31 \( 1 + 6.05T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 2.29iT - 41T^{2} \)
43 \( 1 + 7.23iT - 43T^{2} \)
47 \( 1 + 4.28T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 5.41iT - 61T^{2} \)
67 \( 1 - 3.27iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 14.0iT - 73T^{2} \)
79 \( 1 - 7.91iT - 79T^{2} \)
83 \( 1 + 9.45T + 83T^{2} \)
89 \( 1 - 5.99iT - 89T^{2} \)
97 \( 1 + 9.23iT - 97T^{2} \)
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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

−9.802616016036378047271076421226, −8.944197378494246671009926232749, −8.064050268516588156752844220152, −7.22331541983977326999898993074, −6.49820408562647575838621971633, −5.29768573894946500753116995306, −4.37273078004200382105495532153, −3.48066176705211590264681789042, −2.57714951350536271707406359861, −1.72041227116108906004761835962, 0.22990299786690244910795053257, 1.46536913053903763277420125619, 3.26748081779968369671768653610, 4.46363280900381028774474425170, 4.90603870017765954674529524302, 5.99689083695952244827983885578, 6.45000152021578466506016542140, 7.60204180200485704263079059299, 8.437896720513403660536016107539, 8.745573891363277948397988501848

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