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Properties

Label 2-3648-456.125-c0-0-7
Degree $2$
Conductor $3648$
Sign $0.840 + 0.541i$
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 + 1.73·11-s + (−0.866 + 0.5i)19-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (1.49 − 0.866i)33-s + (1.5 + 0.866i)41-s + (−1.73 − i)43-s − 49-s + (−0.499 + 0.866i)57-s + (0.866 − 1.5i)59-s + (−0.866 + 0.5i)67-s + (0.5 − 0.866i)73-s + (0.866 + 0.499i)75-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.499 − 0.866i)9-s + 1.73·11-s + (−0.866 + 0.5i)19-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (1.49 − 0.866i)33-s + (1.5 + 0.866i)41-s + (−1.73 − i)43-s − 49-s + (−0.499 + 0.866i)57-s + (0.866 − 1.5i)59-s + (−0.866 + 0.5i)67-s + (0.5 − 0.866i)73-s + (0.866 + 0.499i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.840 + 0.541i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ 0.840 + 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.927193456\)
\(L(\frac12)\) \(\approx\) \(1.927193456\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 1.73T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.584654268308063326637447284703, −8.068630335768698583281142302408, −7.06250883951049354947080602081, −6.63684808618431749947897741547, −5.89644652236764756106245799314, −4.64534483487885840530721525383, −3.84253095331439363000355724291, −3.22152514804833536069259133054, −2.02190435125771342779680091526, −1.25405510463981728624149528695, 1.39285572197834043967516395348, 2.43156803345714958681548690246, 3.35523064947248809178457178706, 4.21986417870853949105608461912, 4.63112516001289953972194919901, 5.86629185944619482872825990953, 6.68665169410882831107609197478, 7.27470644508201178405477120284, 8.404719988731342892299331948732, 8.662299257277176805652872483798

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