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Properties

Label 2-3648-456.59-c0-0-2
Degree $2$
Conductor $3648$
Sign $0.971 - 0.236i$
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  + (−0.642 + 0.766i)3-s + (−1.32 + 0.766i)7-s + (−0.173 − 0.984i)9-s + (1.43 − 1.20i)13-s + (0.866 − 0.5i)19-s + (0.266 − 1.50i)21-s + (−0.766 + 0.642i)25-s + (0.866 + 0.500i)27-s + (−0.984 − 1.70i)31-s − 0.347·37-s + 1.87i·39-s + (1.85 − 0.673i)43-s + (0.673 − 1.16i)49-s + (−0.173 + 0.984i)57-s + (−0.233 + 0.642i)61-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)3-s + (−1.32 + 0.766i)7-s + (−0.173 − 0.984i)9-s + (1.43 − 1.20i)13-s + (0.866 − 0.5i)19-s + (0.266 − 1.50i)21-s + (−0.766 + 0.642i)25-s + (0.866 + 0.500i)27-s + (−0.984 − 1.70i)31-s − 0.347·37-s + 1.87i·39-s + (1.85 − 0.673i)43-s + (0.673 − 1.16i)49-s + (−0.173 + 0.984i)57-s + (−0.233 + 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.971 - 0.236i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ 0.971 - 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8589919230\)
\(L(\frac12)\) \(\approx\) \(0.8589919230\)
\(L(1)\) 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 \( 1 + (0.642 - 0.766i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (1.32 - 0.766i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.984 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + 0.347T + T^{2} \)
41 \( 1 + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (-1.85 + 0.673i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.233 - 0.642i)T + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-1.50 + 0.266i)T + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.984 - 0.826i)T + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-1.70 - 0.300i)T + (0.939 + 0.342i)T^{2} \)
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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

−9.064505269660328640949422991978, −8.098179275126308603785808806032, −7.14896433396158188901825102817, −6.19223637774671910354702769445, −5.75701669967590271030506310561, −5.27182399153020446076259198398, −3.84828991238020668377972825080, −3.51815327704964034915516401013, −2.52624906439606882180360318015, −0.72737901158980575407243583007, 0.950599115236907538673685555720, 1.97009339524807957554052129403, 3.33587666705626730338404007984, 3.92107343129491631294701471598, 5.01100102992459820118576173451, 6.09070180916153554347760439000, 6.35727144008757807097610055848, 7.11099514748031309264415018121, 7.71822678008044682023946297466, 8.699141105932810550138523716490

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