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Properties

Label 2-3648-456.149-c0-0-3
Degree $2$
Conductor $3648$
Sign $0.997 + 0.0720i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)9-s + (0.342 − 0.592i)11-s + (−1.11 + 1.32i)17-s + (0.984 − 0.173i)19-s + (0.939 + 0.342i)25-s − 0.999i·27-s + (−0.592 + 0.342i)33-s + (0.673 + 1.85i)41-s + (−0.984 − 0.173i)43-s + (0.5 − 0.866i)49-s + (1.62 − 0.592i)51-s + (−0.939 − 0.342i)57-s + (0.524 + 0.439i)59-s + (0.223 + 0.266i)67-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)9-s + (0.342 − 0.592i)11-s + (−1.11 + 1.32i)17-s + (0.984 − 0.173i)19-s + (0.939 + 0.342i)25-s − 0.999i·27-s + (−0.592 + 0.342i)33-s + (0.673 + 1.85i)41-s + (−0.984 − 0.173i)43-s + (0.5 − 0.866i)49-s + (1.62 − 0.592i)51-s + (−0.939 − 0.342i)57-s + (0.524 + 0.439i)59-s + (0.223 + 0.266i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.997 + 0.0720i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (2657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ 0.997 + 0.0720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9522371242\)
\(L(\frac12)\) \(\approx\) \(0.9522371242\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
good5 \( 1 + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.342 + 0.592i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.173 - 0.984i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.984 + 0.173i)T + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.524 - 0.439i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-0.223 - 0.266i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.984 - 1.70i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
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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

−8.552571326114021782331231712472, −7.995728725716424104823404099731, −7.00503881926737926840125063402, −6.52471621538462648557177174195, −5.80668916198553439292879667139, −5.03931358044783402903362401189, −4.23046817216702119264173958734, −3.21057259347027924704156063722, −2.01308961215611264400873500701, −0.993769636898649907840648988233, 0.819944825955299895910027891348, 2.24200173654252925482537928096, 3.37592372154975557766688400999, 4.30411472880321189180921836437, 4.96189654571108518019341073502, 5.56810670480804053266528198446, 6.67747731057097425055494429603, 6.95030282155563325826365272014, 7.889703624730936610800279467037, 9.176974242950513990202980935718

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