↑

Properties

Label 1-3360-3360.1109-r0-0-0
Degree $1$
Conductor $3360$
Sign $-0.998 - 0.0612i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)11-s + (−0.707 + 0.707i)13-s + (−0.5 − 0.866i)17-s + (0.258 + 0.965i)19-s + (0.866 + 0.5i)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 + (−0.5 + 0.866i)47-s + (0.258 − 0.965i)53-s + (−0.258 + 0.965i)59-s + (−0.258 − 0.965i)61-s + (0.965 + 0.258i)67-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)11-s + (−0.707 + 0.707i)13-s + (−0.5 − 0.866i)17-s + (0.258 + 0.965i)19-s + (0.866 + 0.5i)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 + (−0.5 + 0.866i)47-s + (0.258 − 0.965i)53-s + (−0.258 + 0.965i)59-s + (−0.258 − 0.965i)61-s + (0.965 + 0.258i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.998 - 0.0612i$
Analytic conductor: \(15.6037\)
Root analytic conductor: \(15.6037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3360,\ (0:\ ),\ -0.998 - 0.0612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01120499602 + 0.3654611919i\)
\(L(\frac12)\) \(\approx\) \(0.01120499602 + 0.3654611919i\)
\(L(1)\) \(\approx\) \(0.8297585304 + 0.1504256093i\)
\(L(1)\) \(\approx\) \(0.8297585304 + 0.1504256093i\)

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 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
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 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.258 - 0.965i)T \)
59 \( 1 + (-0.258 + 0.965i)T \)
61 \( 1 + (-0.258 - 0.965i)T \)
67 \( 1 + (0.965 + 0.258i)T \)
71 \( 1 - iT \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.52057024332478573177103432031, −17.74151092073640456905175256886, −17.05625905291431906820330559678, −16.57158276340697223886683735278, −15.50357905737965220307487184705, −15.18081397605629277862542451793, −14.38685594228187768057743698929, −13.3247856320605007996355736112, −13.17874249059861642873022678397, −12.215151461678966486702006598427, −11.33360866852925535502136430956, −10.86592970994995935466949137232, −10.04428632690602027557711358636, −9.2934625646038117198767852647, −8.45470444588375229743611313917, −7.930879364206501746555326623657, −7.00255641255503098439225257188, −6.27776639610315488647757122542, −5.459069188119788612836308455920, −4.78894476280672549292913074791, −3.85669845578543386958511830012, −2.95861016019363127732026707230, −2.35554411799822725599574541843, −1.13470395873622205955412030559, −0.10917900329881189389616662732, 1.44020202562923726959665590776, 2.10649730475505649874010331008, 3.06630326282438080666329271579, 3.90202847911264010318941359632, 4.97422788331964260328992227073, 5.16365564661020876941247396937, 6.46846386534357360708987320379, 7.132723281166258943168587164779, 7.596698081340074782528367830275, 8.70489791542884813758560969695, 9.31885796471115918452951281524, 10.01373663127696237898682378098, 10.69272639207107710094206516825, 11.6521207426698356468695501977, 12.154387834546168072552278368910, 12.8591689507653118013091128519, 13.699030137065211193614876422139, 14.35044334045535665667086828779, 15.01360767091007315662021168211, 15.748065238356212840355609992698, 16.398285617994708710011779098, 17.21196088575168011446729673694, 17.724244554381031416311614674227, 18.550249171100289268273425526788, 19.11056779789293746334882553327

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