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Properties

Label 2-42e2-7.6-c2-0-19
Degree $2$
Conductor $1764$
Sign $0.912 + 0.409i$
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  − 4.25i·5-s + 18.9·11-s − 4.46i·13-s + 12.7i·17-s + 9.37i·19-s + 17.0·23-s + 6.89·25-s − 20.8·29-s − 15.6i·31-s + 45.6·37-s + 63.3i·41-s + 2.20·43-s + 50.6i·47-s − 37.9·53-s − 80.7i·55-s + ⋯
L(s)  = 1  − 0.850i·5-s + 1.72·11-s − 0.343i·13-s + 0.750i·17-s + 0.493i·19-s + 0.742·23-s + 0.275·25-s − 0.720·29-s − 0.503i·31-s + 1.23·37-s + 1.54i·41-s + 0.0511·43-s + 1.07i·47-s − 0.716·53-s − 1.46i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.912 + 0.409i$
Analytic conductor: \(48.0655\)
Root analytic conductor: \(6.93293\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1),\ 0.912 + 0.409i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.358718403\)
\(L(\frac12)\) \(\approx\) \(2.358718403\)
\(L(2)\) 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 \( 1 + 4.25iT - 25T^{2} \)
11 \( 1 - 18.9T + 121T^{2} \)
13 \( 1 + 4.46iT - 169T^{2} \)
17 \( 1 - 12.7iT - 289T^{2} \)
19 \( 1 - 9.37iT - 361T^{2} \)
23 \( 1 - 17.0T + 529T^{2} \)
29 \( 1 + 20.8T + 841T^{2} \)
31 \( 1 + 15.6iT - 961T^{2} \)
37 \( 1 - 45.6T + 1.36e3T^{2} \)
41 \( 1 - 63.3iT - 1.68e3T^{2} \)
43 \( 1 - 2.20T + 1.84e3T^{2} \)
47 \( 1 - 50.6iT - 2.20e3T^{2} \)
53 \( 1 + 37.9T + 2.80e3T^{2} \)
59 \( 1 + 67.6iT - 3.48e3T^{2} \)
61 \( 1 - 63.3iT - 3.72e3T^{2} \)
67 \( 1 - 27.3T + 4.48e3T^{2} \)
71 \( 1 + 15.1T + 5.04e3T^{2} \)
73 \( 1 + 83.8iT - 5.32e3T^{2} \)
79 \( 1 - 133.T + 6.24e3T^{2} \)
83 \( 1 + 109. iT - 6.88e3T^{2} \)
89 \( 1 + 79.9iT - 7.92e3T^{2} \)
97 \( 1 + 119. iT - 9.40e3T^{2} \)
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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

−9.213687722748075220536318434502, −8.320789172794918965157551225856, −7.62084491861487381877946622591, −6.48452189366104981355301369788, −5.96973433036138968713848503273, −4.83400246327860502352157628841, −4.14160827393649181987561665017, −3.21139001673197219738470625858, −1.69993967646611466577145878666, −0.886461995430451802755628795745, 0.901120826635624819666934130976, 2.19000042911596960112195400949, 3.25488686300907784110236395398, 4.05078101850411330703071681376, 5.05793722873373843037704832251, 6.14649120337170693140555479345, 6.92652501060716403333314040287, 7.22750582264852868496079560987, 8.537666665991459719801532233098, 9.232383001746600910880498101213

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