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Properties

Label 1-3648-3648.59-r1-0-0
Degree $1$
Conductor $3648$
Sign $0.969 + 0.246i$
Analytic cond. $392.032$
Root an. cond. $392.032$
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  + (−0.953 + 0.300i)5-s + (−0.965 + 0.258i)7-s + (0.991 − 0.130i)11-s + (0.216 + 0.976i)13-s + (0.342 − 0.939i)17-s + (0.996 − 0.0871i)23-s + (0.819 − 0.573i)25-s + (0.675 + 0.737i)29-s + (0.5 − 0.866i)31-s + (0.843 − 0.537i)35-s + (0.923 + 0.382i)37-s + (0.819 + 0.573i)41-s + (0.300 + 0.953i)43-s + (0.342 + 0.939i)47-s + (0.866 − 0.5i)49-s + ⋯
L(s)  = 1  + (−0.953 + 0.300i)5-s + (−0.965 + 0.258i)7-s + (0.991 − 0.130i)11-s + (0.216 + 0.976i)13-s + (0.342 − 0.939i)17-s + (0.996 − 0.0871i)23-s + (0.819 − 0.573i)25-s + (0.675 + 0.737i)29-s + (0.5 − 0.866i)31-s + (0.843 − 0.537i)35-s + (0.923 + 0.382i)37-s + (0.819 + 0.573i)41-s + (0.300 + 0.953i)43-s + (0.342 + 0.939i)47-s + (0.866 − 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(392.032\)
Root analytic conductor: \(392.032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3648,\ (1:\ ),\ 0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.930845740 + 0.2421698954i\)
\(L(\frac12)\) \(\approx\) \(1.930845740 + 0.2421698954i\)
\(L(1)\) \(\approx\) \(0.9622904525 + 0.09680083523i\)
\(L(1)\) \(\approx\) \(0.9622904525 + 0.09680083523i\)

Euler product

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

−18.783233208640115749970388160512, −17.49157897081088894671040502172, −17.187926741796935619155346103327, −16.31660993651017830911834224180, −15.76053325884011173960425933111, −15.14354568404887356955842461118, −14.46549193838000411442616133652, −13.54284463873596595393053586146, −12.71685559171395043131358316081, −12.43055547021903763266497626030, −11.58721590383056787446952551482, −10.78782278773238684238649688120, −10.15045607884084135276114817342, −9.295625199164274896449924305660, −8.61085372912439625721952275065, −7.92088973823813680672477970888, −7.1117849791603902674374399871, −6.46865619573950820226624285909, −5.66818912240370188784652651790, −4.71084879775661110458474114934, −3.79710201786187012844372935107, −3.49118456663816188263313992061, −2.50404813015327514891135774707, −1.096057079892310702954505215225, −0.62739626426036564989825471423, 0.54794502567384288084078306588, 1.33550435311718772570498521474, 2.847341656772955616552311769442, 3.053540672383007984071847773227, 4.21738117900658928478930172045, 4.55118794400667328175309497743, 5.8997553758436576402727187612, 6.5257840204672362804098761626, 7.10846522301684831790231237785, 7.84556815350811133504985389594, 8.91719785096189649330400825478, 9.25404151255608351776717346074, 10.06742956784010565827365444639, 11.13492146207134035000663808632, 11.54355046141636546440727663681, 12.21183832845555656434269335079, 12.893305023455006542089665338923, 13.78372820572788098847196038145, 14.47046187346556488075065924875, 15.07244171306061959797307295818, 15.94345805564708692129698957782, 16.37102483537522044709975756227, 16.90443214830939061905839839366, 17.99728198610637247192359446494, 18.73073759124147437691327713367

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