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Properties

Label 1-3360-3360.227-r0-0-0
Degree $1$
Conductor $3360$
Sign $0.220 - 0.975i$
Analytic cond. $15.6037$
Root an. cond. $15.6037$
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  + (−0.965 − 0.258i)11-s + (0.707 − 0.707i)13-s + (−0.866 + 0.5i)17-s + (−0.965 + 0.258i)19-s + (−0.5 + 0.866i)23-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (0.965 − 0.258i)37-s + i·41-s + (0.707 + 0.707i)43-s + (−0.866 − 0.5i)47-s + (0.258 − 0.965i)53-s + (0.965 + 0.258i)59-s + (0.965 − 0.258i)61-s + (−0.965 − 0.258i)67-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)11-s + (0.707 − 0.707i)13-s + (−0.866 + 0.5i)17-s + (−0.965 + 0.258i)19-s + (−0.5 + 0.866i)23-s + (0.707 + 0.707i)29-s + (−0.5 − 0.866i)31-s + (0.965 − 0.258i)37-s + i·41-s + (0.707 + 0.707i)43-s + (−0.866 − 0.5i)47-s + (0.258 − 0.965i)53-s + (0.965 + 0.258i)59-s + (0.965 − 0.258i)61-s + (−0.965 − 0.258i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(15.6037\)
Root analytic conductor: \(15.6037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3360,\ (0:\ ),\ 0.220 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8267652915 - 0.6608284570i\)
\(L(\frac12)\) \(\approx\) \(0.8267652915 - 0.6608284570i\)
\(L(1)\) \(\approx\) \(0.9140876017 - 0.08152725725i\)
\(L(1)\) \(\approx\) \(0.9140876017 - 0.08152725725i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (-0.965 - 0.258i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.965 - 0.258i)T \)
41 \( 1 + iT \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.258 - 0.965i)T \)
59 \( 1 + (0.965 + 0.258i)T \)
61 \( 1 + (0.965 - 0.258i)T \)
67 \( 1 + (-0.965 - 0.258i)T \)
71 \( 1 - iT \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 - iT \)
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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

−18.96924379204296641378663027114, −18.1959821365012832216018630484, −17.77519368709127721415783892662, −16.883116370151565890750378513992, −16.065135225273006850401015465441, −15.69130908409347326498902399954, −14.85571072527034092491519938755, −14.056520912603699931916504423894, −13.42312547225942473701571953716, −12.76868696587477681890007614486, −12.039463210757463605797298292497, −11.14989659334865253931089111363, −10.64433978000957210858189773116, −9.89386908512860781419421568713, −8.93081831136605306910583187317, −8.47561242251415491986200806624, −7.59561578866836914153287324532, −6.732702968777338384869355229542, −6.20469627657692706849990149420, −5.19266710923779703215258826907, −4.45083509288695815763837140671, −3.82006368900059178348317135720, −2.52989359157581079234596760945, −2.20386760940584263894602853264, −0.884339680123842862785979165075, 0.36545253562061950342433403814, 1.59960690819912488916595307050, 2.43780048428621597316886584606, 3.30969199233592014455397178646, 4.104152596434434508946535899271, 4.95059561562919200131884452402, 5.84966541998621900347294743273, 6.31340394309918485067586440257, 7.36239735281907120197756955963, 8.17510454409184618530914704615, 8.535714420661844906404113637716, 9.59112002493379363119282721054, 10.31489278500882632447447624611, 10.98620747148442508424223107218, 11.49500844697840888643381251724, 12.68693329206472032842973479826, 13.06756234591023163915436379327, 13.62953830288766635570387206318, 14.730721019768699667016079323622, 15.15119850215196935088610718416, 16.037608682323557911950137970780, 16.40616335287302973945231400117, 17.51143477301836704912362211552, 17.97077354066693413726222442259, 18.55183490982100722821235450300

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