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Properties

Label 1-3648-3648.1229-r0-0-0
Degree $1$
Conductor $3648$
Sign $-0.715 - 0.698i$
Analytic cond. $16.9412$
Root an. cond. $16.9412$
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  + (0.737 − 0.675i)5-s + (0.965 + 0.258i)7-s + (−0.991 − 0.130i)11-s + (−0.953 + 0.300i)13-s + (−0.984 − 0.173i)17-s + (0.422 + 0.906i)23-s + (0.0871 − 0.996i)25-s + (−0.976 − 0.216i)29-s + (−0.5 − 0.866i)31-s + (0.887 − 0.461i)35-s + (0.923 − 0.382i)37-s + (0.0871 + 0.996i)41-s + (−0.675 − 0.737i)43-s + (0.984 − 0.173i)47-s + (0.866 + 0.5i)49-s + ⋯
L(s)  = 1  + (0.737 − 0.675i)5-s + (0.965 + 0.258i)7-s + (−0.991 − 0.130i)11-s + (−0.953 + 0.300i)13-s + (−0.984 − 0.173i)17-s + (0.422 + 0.906i)23-s + (0.0871 − 0.996i)25-s + (−0.976 − 0.216i)29-s + (−0.5 − 0.866i)31-s + (0.887 − 0.461i)35-s + (0.923 − 0.382i)37-s + (0.0871 + 0.996i)41-s + (−0.675 − 0.737i)43-s + (0.984 − 0.173i)47-s + (0.866 + 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $-0.715 - 0.698i$
Analytic conductor: \(16.9412\)
Root analytic conductor: \(16.9412\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (1229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3648,\ (0:\ ),\ -0.715 - 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3635840341 - 0.8923240850i\)
\(L(\frac12)\) \(\approx\) \(0.3635840341 - 0.8923240850i\)
\(L(1)\) \(\approx\) \(1.004727344 - 0.1977839860i\)
\(L(1)\) \(\approx\) \(1.004727344 - 0.1977839860i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.737 - 0.675i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
11 \( 1 + (-0.991 - 0.130i)T \)
13 \( 1 + (-0.953 + 0.300i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.422 + 0.906i)T \)
29 \( 1 + (-0.976 - 0.216i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (0.0871 + 0.996i)T \)
43 \( 1 + (-0.675 - 0.737i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 + (0.0436 + 0.999i)T \)
59 \( 1 + (-0.216 - 0.976i)T \)
61 \( 1 + (-0.675 + 0.737i)T \)
67 \( 1 + (-0.976 - 0.216i)T \)
71 \( 1 + (0.422 - 0.906i)T \)
73 \( 1 + (-0.0871 - 0.996i)T \)
79 \( 1 + (-0.642 + 0.766i)T \)
83 \( 1 + (-0.130 - 0.991i)T \)
89 \( 1 + (0.0871 - 0.996i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.55798319168866418678085997337, −18.342140517550945487215420668330, −17.49615986815584963543509776214, −17.1632528266997281078521281990, −16.22462462938623761943943295857, −15.24716758326656268971453791043, −14.77790288637314914204265257960, −14.25600419247452064345849036681, −13.35059204753801970372190824842, −12.910956786800154487432917917, −11.97689271491824265180139729301, −10.96572961560808912945841298524, −10.71797641986824859498347771278, −9.99680728169981163996841169126, −9.14672244747328699515158438993, −8.35431840718020748813533761887, −7.488797736548230107471205171444, −7.04041327979556564938304577111, −6.10634635845657881812311414971, −5.20162437769781649289042101534, −4.80944277204325303660235594239, −3.751429699982888996843371816523, −2.55882370038298652729632082865, −2.31398441720162781185488961851, −1.2511009537173309744210167162, 0.24908903584091479371726382860, 1.58393307853124307276074419753, 2.139840710305091094027834006774, 2.88074539647918520511632510984, 4.28303803362092522144723894777, 4.78954113381300958128454673232, 5.50230374013739899864213590152, 6.047677186273937410981148433751, 7.32465640942344399619528989133, 7.72913326163857061201824196072, 8.69875889805124821048228798046, 9.24779702727163624877194340198, 9.91447661355774108921360364757, 10.842524438935633533544974430941, 11.41540318051819381437000849149, 12.20901659237962218458311817640, 13.02867716904533897077048654259, 13.48379052863770337821796123168, 14.21325811313516783390367337693, 15.11625945857587311177799607886, 15.468143347886714387406786247543, 16.64093917710924789320615819813, 16.95269219997631606009543299158, 17.810553705786736218962166777894, 18.211605268032370124401983237746

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