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Properties

Label 1-60e2-3600.1219-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.00174 - 0.999i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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  + (0.5 − 0.866i)7-s + (0.994 − 0.104i)11-s + (0.994 + 0.104i)13-s + (−0.309 − 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.913 − 0.406i)23-s + (−0.743 + 0.669i)29-s + (−0.669 + 0.743i)31-s + (0.587 − 0.809i)37-s + (0.104 − 0.994i)41-s + (0.866 + 0.5i)43-s + (0.669 + 0.743i)47-s + (−0.5 − 0.866i)49-s + (0.951 + 0.309i)53-s + (−0.994 − 0.104i)59-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)7-s + (0.994 − 0.104i)11-s + (0.994 + 0.104i)13-s + (−0.309 − 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.913 − 0.406i)23-s + (−0.743 + 0.669i)29-s + (−0.669 + 0.743i)31-s + (0.587 − 0.809i)37-s + (0.104 − 0.994i)41-s + (0.866 + 0.5i)43-s + (0.669 + 0.743i)47-s + (−0.5 − 0.866i)49-s + (0.951 + 0.309i)53-s + (−0.994 − 0.104i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00174 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00174 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.00174 - 0.999i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.00174 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.055195744 - 2.051611878i\)
\(L(\frac12)\) \(\approx\) \(2.055195744 - 2.051611878i\)
\(L(1)\) \(\approx\) \(1.267711839 - 0.3008917107i\)
\(L(1)\) \(\approx\) \(1.267711839 - 0.3008917107i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.994 - 0.104i)T \)
13 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.951 - 0.309i)T \)
23 \( 1 + (-0.913 - 0.406i)T \)
29 \( 1 + (-0.743 + 0.669i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (0.104 - 0.994i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (-0.994 - 0.104i)T \)
61 \( 1 + (0.994 - 0.104i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.669 - 0.743i)T \)
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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

−18.48785497060445698013864624272, −18.19439063795564691842061555696, −17.30786428916890155396190717345, −16.717887010479817244812602507974, −15.83720884838296011974614615430, −15.25001434324309800521795776730, −14.64169910040544338951806097355, −13.90656128867276689807638626621, −13.183047701493102713296195252043, −12.39922273493785395422145086405, −11.53997300350111453709248835014, −11.37903702160263577699660787609, −10.27758384051327401027579052397, −9.47848342527537983897972468753, −8.873976251232056377078923310618, −8.14729742896886654261798601900, −7.52335163846457786576206556513, −6.35169870085257024968030499993, −5.93619772670203327417967787675, −5.20453740328591689382359450428, −4.012647388967059069695011219570, −3.717447831651118545423907213674, −2.45627931864698580020145980150, −1.71307154146285983566063438609, −0.96483222001698649785626017888, 0.49768485428731414979605479981, 1.184052932542579484263082413753, 2.0199711337688316132186925414, 3.20036836219931031045435489726, 3.92089069516656032121242625859, 4.5103676804607847691600571859, 5.49590454230575087673322445436, 6.224601165259942163055703254942, 7.21970494771374506011546090894, 7.46883947964958375950094053292, 8.663424034831429685905887357858, 9.10669001142995943305094354677, 9.946146236287612615382293664728, 10.896355597661906832792138329823, 11.25066195196798676440963269452, 12.009529023324646059923948150723, 12.86393732065525790539361127349, 13.75421333828640170498160508407, 14.12545445465329512351832623215, 14.68255680488451404179302960421, 15.9325317050436682086445974692, 16.138847651940035680538112156560, 16.98582748569761647288052369240, 17.777836610119111534626947058997, 18.17911878535297100464725243002

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