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Properties

Label 1-1620-1620.959-r0-0-0
Degree $1$
Conductor $1620$
Sign $0.305 - 0.952i$
Analytic cond. $7.52324$
Root an. cond. $7.52324$
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  + (−0.686 + 0.727i)7-s + (0.893 − 0.448i)11-s + (−0.396 − 0.918i)13-s + (0.766 − 0.642i)17-s + (−0.766 − 0.642i)19-s + (0.686 + 0.727i)23-s + (0.993 − 0.116i)29-s + (−0.973 + 0.230i)31-s + (−0.173 − 0.984i)37-s + (−0.597 + 0.802i)41-s + (−0.0581 + 0.998i)43-s + (−0.973 − 0.230i)47-s + (−0.0581 − 0.998i)49-s + (−0.5 − 0.866i)53-s + (0.893 + 0.448i)59-s + ⋯
L(s)  = 1  + (−0.686 + 0.727i)7-s + (0.893 − 0.448i)11-s + (−0.396 − 0.918i)13-s + (0.766 − 0.642i)17-s + (−0.766 − 0.642i)19-s + (0.686 + 0.727i)23-s + (0.993 − 0.116i)29-s + (−0.973 + 0.230i)31-s + (−0.173 − 0.984i)37-s + (−0.597 + 0.802i)41-s + (−0.0581 + 0.998i)43-s + (−0.973 − 0.230i)47-s + (−0.0581 − 0.998i)49-s + (−0.5 − 0.866i)53-s + (0.893 + 0.448i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(7.52324\)
Root analytic conductor: \(7.52324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1620,\ (0:\ ),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9612582040 - 0.7012471165i\)
\(L(\frac12)\) \(\approx\) \(0.9612582040 - 0.7012471165i\)
\(L(1)\) \(\approx\) \(0.9685905420 - 0.1176108764i\)
\(L(1)\) \(\approx\) \(0.9685905420 - 0.1176108764i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.893 - 0.448i)T \)
13 \( 1 + (-0.396 - 0.918i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.686 + 0.727i)T \)
29 \( 1 + (0.993 - 0.116i)T \)
31 \( 1 + (-0.973 + 0.230i)T \)
37 \( 1 + (-0.173 - 0.984i)T \)
41 \( 1 + (-0.597 + 0.802i)T \)
43 \( 1 + (-0.0581 + 0.998i)T \)
47 \( 1 + (-0.973 - 0.230i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.893 + 0.448i)T \)
61 \( 1 + (-0.286 - 0.957i)T \)
67 \( 1 + (-0.993 - 0.116i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (-0.597 - 0.802i)T \)
83 \( 1 + (-0.597 - 0.802i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.835 + 0.549i)T \)
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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

−20.5783571371596513453783303915, −19.737605419710324867385628684155, −19.17577571450397567301847697783, −18.58149967072040135522968689053, −17.31827845999781205473768010741, −16.87539832644933832158831243706, −16.39714942608960582484296333787, −15.27822256853208307789076812535, −14.48903707860201928940351418191, −14.014875003360556908412067766666, −12.94762224410798462175696148499, −12.36256273239595463063718228264, −11.621136878195312540527112928427, −10.50562902235029328926271282870, −10.04549412742515557149940959681, −9.15023121539671674353672542503, −8.39849124329876859244362881559, −7.2551395538809015148948014624, −6.72720188629317504345946240535, −5.990539171495761816604912107438, −4.730749755205480164367598442660, −4.02749943040506722069781528965, −3.2817958103525166657650565057, −2.03832373311073491564361852435, −1.12406034261518045417311746960, 0.47690747727972846509729706725, 1.74569958985166761046048146535, 2.98637475805923295716590027418, 3.35433139755729992974456598288, 4.69555534172713187564204122520, 5.490783097371132234299396228127, 6.29487198734034850782774624677, 7.05109584227106829280080753033, 8.03513448685890593988681728987, 8.92980318422897695117965018390, 9.49946960478440215627245467513, 10.32783226145208238456360852713, 11.32232680087795054428951669281, 11.97830205825842250385781559199, 12.80391421863063815746713412213, 13.36464755442202984236655158864, 14.5130521474343982680550224302, 14.94937730277569629923734720936, 15.92695344900480744836463534643, 16.45198039673369777290231683074, 17.39044721532929634709298239516, 18.017999197696250352141989374571, 18.97040068234793247635298374170, 19.53253883289377630153833079252, 20.05982831223123167353522736952

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