↑

Properties

Label 2-3360-5.4-c1-0-29
Degree $2$
Conductor $3360$
Sign $0.894 - 0.447i$
Analytic cond. $26.8297$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + (1 + 2i)5-s − i·7-s − 9-s − 2·11-s − 2i·13-s + (2 − i)15-s + 6·19-s − 21-s + 8i·23-s + (−3 + 4i)25-s + i·27-s − 6·29-s + 10·31-s + 2i·33-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.447 + 0.894i)5-s − 0.377i·7-s − 0.333·9-s − 0.603·11-s − 0.554i·13-s + (0.516 − 0.258i)15-s + 1.37·19-s − 0.218·21-s + 1.66i·23-s + (−0.600 + 0.800i)25-s + 0.192i·27-s − 1.11·29-s + 1.79·31-s + 0.348i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.853784117\)
\(L(\frac12)\) \(\approx\) \(1.853784117\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + (-1 - 2i)T \)
7 \( 1 + iT \)
good11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.463480081959772976007979973459, −7.78224271891137787612796157022, −7.17817472778533811535081075798, −6.60973848332088299689532317438, −5.55660039696811541272041611687, −5.23127316502946779644132125279, −3.68941056449865305864174554438, −3.08034955719380830014662360785, −2.14390266808538607461283957311, −1.01718630096362437117326236304, 0.66388377888236894178438212334, 2.03282432851473089476239779270, 2.90208718495021697407384421140, 4.05937509446663104239091134499, 4.80914988616471396835310900793, 5.40097187658436340098874843807, 6.12179750475866561004483171454, 7.04974329124393964443565938055, 8.138637811203088928535621409376, 8.530681493647580361261043034161

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