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Properties

Label 1-3648-3648.1661-r0-0-0
Degree $1$
Conductor $3648$
Sign $0.258 - 0.965i$
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  + (−0.608 + 0.793i)5-s + (0.707 − 0.707i)7-s + (−0.923 + 0.382i)11-s + (−0.608 − 0.793i)13-s + (−0.866 + 0.5i)17-s + (−0.965 + 0.258i)23-s + (−0.258 − 0.965i)25-s + (−0.793 + 0.608i)29-s + 31-s + (0.130 + 0.991i)35-s + (0.382 + 0.923i)37-s + (−0.258 + 0.965i)41-s + (0.793 + 0.608i)43-s + (0.866 + 0.5i)47-s − i·49-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)5-s + (0.707 − 0.707i)7-s + (−0.923 + 0.382i)11-s + (−0.608 − 0.793i)13-s + (−0.866 + 0.5i)17-s + (−0.965 + 0.258i)23-s + (−0.258 − 0.965i)25-s + (−0.793 + 0.608i)29-s + 31-s + (0.130 + 0.991i)35-s + (0.382 + 0.923i)37-s + (−0.258 + 0.965i)41-s + (0.793 + 0.608i)43-s + (0.866 + 0.5i)47-s − i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(16.9412\)
Root analytic conductor: \(16.9412\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (1661, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3648,\ (0:\ ),\ 0.258 - 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5912981052 - 0.4538516863i\)
\(L(\frac12)\) \(\approx\) \(0.5912981052 - 0.4538516863i\)
\(L(1)\) \(\approx\) \(0.8153037554 + 0.01979334673i\)
\(L(1)\) \(\approx\) \(0.8153037554 + 0.01979334673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 + (-0.608 - 0.793i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.965 + 0.258i)T \)
29 \( 1 + (-0.793 + 0.608i)T \)
31 \( 1 + T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.258 + 0.965i)T \)
43 \( 1 + (0.793 + 0.608i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (-0.130 + 0.991i)T \)
59 \( 1 + (0.608 - 0.793i)T \)
61 \( 1 + (0.793 - 0.608i)T \)
67 \( 1 + (-0.793 + 0.608i)T \)
71 \( 1 + (-0.965 - 0.258i)T \)
73 \( 1 + (0.258 - 0.965i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (-0.258 - 0.965i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−18.957318021180393625529329971915, −18.06213978060794410888925077144, −17.52525010998378187560213434104, −16.64285151701178526442237326569, −16.00620812753151057210173428751, −15.49154410033079929846837297360, −14.82128235062380214119342328833, −13.904651087507608628243857638695, −13.33833790532740919201117447, −12.404126930267383388448286254168, −11.9187348486690815668732343272, −11.3381148558046920323298568756, −10.569156965106300065824922881630, −9.5335789214361131668825945420, −8.913427747011237311513765128561, −8.27775166080106634384410549035, −7.67497665958058875059930737630, −6.88709544214286104863080609156, −5.71258820048678207181209918074, −5.25307227682283395781257318753, −4.410237857622806700036968682569, −3.887067228495816509835842151126, −2.421375269786600578600448548449, −2.175770569191249541544263807136, −0.81194748191433802384311671821, 0.26635349657720839294393403356, 1.58611863831686585323074598988, 2.52379913869886913442458726264, 3.1935489656177761312798412246, 4.24719904216675226821106853931, 4.66233176335824998888526425785, 5.68364600774657318195936443588, 6.553437015801389798321508753779, 7.400148652071716382647453833965, 7.84214918091178532239802257734, 8.36675749615918415123428431963, 9.63426060473663759321398738557, 10.36388970959204930037908140483, 10.76990932452346714120299956018, 11.481386088899979378684785051746, 12.23800979870282217477996959067, 13.07042693655335771292186054958, 13.70421878641238208558888214857, 14.57855528878803083120581933220, 15.05112990625049858288824728390, 15.644578653787898431065392186818, 16.38561298890200844159451238579, 17.50477812337384257590995313611, 17.68330311062466487854258180665, 18.470335225019758568779950974211

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