Properties

Label 2-42e2-7.3-c2-0-9
Degree $2$
Conductor $1764$
Sign $-0.379 - 0.925i$
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  + (4.65 − 2.68i)5-s + (−4.29 + 7.44i)11-s + 21.0i·13-s + (4.75 + 2.74i)17-s + (6.27 − 3.62i)19-s + (14.0 + 24.2i)23-s + (1.97 − 3.41i)25-s − 40.3·29-s + (−35.0 − 20.2i)31-s + (−33.3 − 57.7i)37-s + 33.6i·41-s + 0.932·43-s + (−74.1 + 42.8i)47-s + (−22.2 + 38.6i)53-s + 46.2i·55-s + ⋯
L(s)  = 1  + (0.931 − 0.537i)5-s + (−0.390 + 0.676i)11-s + 1.61i·13-s + (0.279 + 0.161i)17-s + (0.330 − 0.190i)19-s + (0.609 + 1.05i)23-s + (0.0788 − 0.136i)25-s − 1.39·29-s + (−1.13 − 0.653i)31-s + (−0.900 − 1.55i)37-s + 0.820i·41-s + 0.0216·43-s + (−1.57 + 0.911i)47-s + (−0.420 + 0.728i)53-s + 0.840i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.379 - 0.925i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ -0.379 - 0.925i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.426252475\)
\(L(\frac12)\) \(\approx\) \(1.426252475\)
\(L(2)\) 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 \( 1 + (-4.65 + 2.68i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (4.29 - 7.44i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 21.0iT - 169T^{2} \)
17 \( 1 + (-4.75 - 2.74i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.27 + 3.62i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-14.0 - 24.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 40.3T + 841T^{2} \)
31 \( 1 + (35.0 + 20.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (33.3 + 57.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 33.6iT - 1.68e3T^{2} \)
43 \( 1 - 0.932T + 1.84e3T^{2} \)
47 \( 1 + (74.1 - 42.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (22.2 - 38.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-55.1 - 31.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (27.7 - 16.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-23.8 + 41.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 14.9T + 5.04e3T^{2} \)
73 \( 1 + (121. + 70.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-61.1 - 105. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 33.1iT - 6.88e3T^{2} \)
89 \( 1 + (-31.2 + 18.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 16.2iT - 9.40e3T^{2} \)
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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

−9.422270559342611609089386725279, −8.884041265028060811190437824790, −7.59986488861228317199896514913, −7.10405450498912127782532135955, −6.00063040070389993067053813505, −5.35257597652306923237072973448, −4.52101107208905830477114210046, −3.50637123989116143978283086239, −2.07223597512537726171794720632, −1.52122248754058929721941656438, 0.34279091102680738577864565646, 1.76323793966743518882723060316, 2.91907021492625223860947896541, 3.48272968011168813388718192607, 5.11905391241782250011171405180, 5.53586083636137311109853867624, 6.40614521332692313035615890421, 7.24091401690544185782695791142, 8.142985627431659548544808711416, 8.808187148083001183601770625865

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