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Properties

Label 1-3648-3648.1853-r0-0-0
Degree $1$
Conductor $3648$
Sign $-0.510 + 0.859i$
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  + (0.300 − 0.953i)5-s + (−0.965 − 0.258i)7-s + (0.130 − 0.991i)11-s + (−0.976 − 0.216i)13-s + (0.342 + 0.939i)17-s + (0.996 + 0.0871i)23-s + (−0.819 − 0.573i)25-s + (−0.737 − 0.675i)29-s + (−0.5 − 0.866i)31-s + (−0.537 + 0.843i)35-s + (0.382 + 0.923i)37-s + (−0.819 + 0.573i)41-s + (−0.953 − 0.300i)43-s + (−0.342 + 0.939i)47-s + (0.866 + 0.5i)49-s + ⋯
L(s)  = 1  + (0.300 − 0.953i)5-s + (−0.965 − 0.258i)7-s + (0.130 − 0.991i)11-s + (−0.976 − 0.216i)13-s + (0.342 + 0.939i)17-s + (0.996 + 0.0871i)23-s + (−0.819 − 0.573i)25-s + (−0.737 − 0.675i)29-s + (−0.5 − 0.866i)31-s + (−0.537 + 0.843i)35-s + (0.382 + 0.923i)37-s + (−0.819 + 0.573i)41-s + (−0.953 − 0.300i)43-s + (−0.342 + 0.939i)47-s + (0.866 + 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $-0.510 + 0.859i$
Analytic conductor: \(16.9412\)
Root analytic conductor: \(16.9412\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (1853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3648,\ (0:\ ),\ -0.510 + 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03095843135 - 0.05438877440i\)
\(L(\frac12)\) \(\approx\) \(0.03095843135 - 0.05438877440i\)
\(L(1)\) \(\approx\) \(0.7695754931 - 0.2309161119i\)
\(L(1)\) \(\approx\) \(0.7695754931 - 0.2309161119i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.300 - 0.953i)T \)
7 \( 1 + (-0.965 - 0.258i)T \)
11 \( 1 + (0.130 - 0.991i)T \)
13 \( 1 + (-0.976 - 0.216i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (0.996 + 0.0871i)T \)
29 \( 1 + (-0.737 - 0.675i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.819 + 0.573i)T \)
43 \( 1 + (-0.953 - 0.300i)T \)
47 \( 1 + (-0.342 + 0.939i)T \)
53 \( 1 + (0.461 - 0.887i)T \)
59 \( 1 + (-0.675 - 0.737i)T \)
61 \( 1 + (-0.953 + 0.300i)T \)
67 \( 1 + (-0.737 - 0.675i)T \)
71 \( 1 + (0.996 - 0.0871i)T \)
73 \( 1 + (0.819 - 0.573i)T \)
79 \( 1 + (0.984 + 0.173i)T \)
83 \( 1 + (-0.991 + 0.130i)T \)
89 \( 1 + (-0.819 - 0.573i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−19.13010950109183032551155917313, −18.24846175235939240646084368412, −18.07960755190418962722212952972, −16.84462390252188510383793237896, −16.648468752947619030834610654471, −15.41840932881459619230843435899, −15.11594941980071726539847668389, −14.34184801379074363089398337436, −13.71552772936480534515804079, −12.79476748548429114705479426385, −12.30364651829231862096551978685, −11.52476304396917721697734125754, −10.65907302703738152323676904613, −9.97030800859350721977722601399, −9.47518327757362664404905531091, −8.85249428272038466121808616217, −7.41429906805716559643756430022, −7.14003583689695722939536842243, −6.56866167593221052835713365664, −5.524707548388712212471293397, −4.94517656882198594254615270448, −3.821005567617675836417830955863, −3.03364787925457338309652631801, −2.471083987846770253339130169278, −1.57966175799842701435595462145, 0.0192140031573286131361909488, 0.98257439562483830818265271965, 1.93857738897787828798017275783, 3.01436928223484484851107588094, 3.65178435029322745578877864855, 4.57914887198885294795282088042, 5.37267804566482778998033249403, 6.05556942609576119503131944392, 6.706611500174719493679176728067, 7.78185676228704305773916992622, 8.309316767197910395549299169723, 9.290612631366287391959644439492, 9.63806507382759934587727543224, 10.42940304438432805667680125606, 11.30107957687844059841722065510, 12.1075358495670989091113395128, 12.83091791763561480453603063378, 13.254422314534370247050765514226, 13.87446241478261291013023418233, 14.92940150073607948312233373670, 15.41557645616680187038841042377, 16.46815636545119222598436374134, 16.911272935949546311987303463085, 17.066790643453290272418140308635, 18.297614506311788705386238492494

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