Properties

Label 2-42e2-21.11-c2-0-7
Degree $2$
Conductor $1764$
Sign $-0.807 - 0.590i$
Analytic cond. $48.0655$
Root an. cond. $6.93293$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.27 + 4.77i)5-s + (8.03 + 4.63i)11-s − 4.24·13-s + (13.4 + 7.77i)17-s + (17.4 + 30.2i)19-s + (15.3 − 8.87i)23-s + (33.1 − 57.4i)25-s + 26.2i·29-s + (16.0 − 27.7i)31-s + (−27.6 − 47.9i)37-s + 38.4i·41-s + 29.3·43-s + (−59.0 + 34.1i)47-s + (−0.943 − 0.544i)53-s − 88.6·55-s + ⋯
L(s)  = 1  + (−1.65 + 0.955i)5-s + (0.730 + 0.421i)11-s − 0.326·13-s + (0.792 + 0.457i)17-s + (0.918 + 1.59i)19-s + (0.668 − 0.386i)23-s + (1.32 − 2.29i)25-s + 0.904i·29-s + (0.517 − 0.895i)31-s + (−0.747 − 1.29i)37-s + 0.937i·41-s + 0.682·43-s + (−1.25 + 0.725i)47-s + (−0.0177 − 0.0102i)53-s − 1.61·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.807 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.807 - 0.590i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.807 - 0.590i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.069100334\)
\(L(\frac12)\) \(\approx\) \(1.069100334\)
\(L(2)\) 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 \( 1 + (8.27 - 4.77i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-8.03 - 4.63i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 4.24T + 169T^{2} \)
17 \( 1 + (-13.4 - 7.77i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-17.4 - 30.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-15.3 + 8.87i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 26.2iT - 841T^{2} \)
31 \( 1 + (-16.0 + 27.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (27.6 + 47.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 38.4iT - 1.68e3T^{2} \)
43 \( 1 - 29.3T + 1.84e3T^{2} \)
47 \( 1 + (59.0 - 34.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (0.943 + 0.544i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-59.0 - 34.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (23.8 + 41.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-4.32 + 7.49i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 95.7iT - 5.04e3T^{2} \)
73 \( 1 + (45.0 - 77.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-74.3 - 128. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 88.8iT - 6.88e3T^{2} \)
89 \( 1 + (-66.0 + 38.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 12.7T + 9.40e3T^{2} \)
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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

−9.527869189948759173155084857185, −8.359274193275321773153569341044, −7.80515098465885344498120949828, −7.16524602939868119122885760994, −6.46232006030298150858790855863, −5.34772792674953791828361756700, −4.14569345099500486139387185313, −3.66033879411052774569807764576, −2.76788924758424428127487795231, −1.18644191708580501999894605701, 0.36382607788023065819112502747, 1.17639515570394416088143326419, 3.05125139374693307041156923437, 3.69340345506617181274129981936, 4.81355975445102007145538778735, 5.13148929004254157539992520120, 6.60083120122064521351992710441, 7.38838480669959621449477203430, 7.950727923712442924943883871703, 8.903557128578838664787948223897

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