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Properties

Label 2-3360-40.29-c1-0-4
Degree $2$
Conductor $3360$
Sign $-0.624 + 0.780i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (−0.836 + 2.07i)5-s + i·7-s + 9-s + 5.53i·11-s − 5.75·13-s + (0.836 − 2.07i)15-s − 0.947i·17-s + 8.10i·19-s − i·21-s + 1.91i·23-s + (−3.60 − 3.46i)25-s − 27-s − 2.56i·29-s − 7.22·31-s + ⋯
L(s)  = 1  − 0.577·3-s + (−0.374 + 0.927i)5-s + 0.377i·7-s + 0.333·9-s + 1.67i·11-s − 1.59·13-s + (0.215 − 0.535i)15-s − 0.229i·17-s + 1.85i·19-s − 0.218i·21-s + 0.398i·23-s + (−0.720 − 0.693i)25-s − 0.192·27-s − 0.476i·29-s − 1.29·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.624 + 0.780i$
Analytic conductor: \(26.8297\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ -0.624 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4138862506\)
\(L(\frac12)\) \(\approx\) \(0.4138862506\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + (0.836 - 2.07i)T \)
7 \( 1 - iT \)
good11 \( 1 - 5.53iT - 11T^{2} \)
13 \( 1 + 5.75T + 13T^{2} \)
17 \( 1 + 0.947iT - 17T^{2} \)
19 \( 1 - 8.10iT - 19T^{2} \)
23 \( 1 - 1.91iT - 23T^{2} \)
29 \( 1 + 2.56iT - 29T^{2} \)
31 \( 1 + 7.22T + 31T^{2} \)
37 \( 1 + 0.366T + 37T^{2} \)
41 \( 1 - 1.45T + 41T^{2} \)
43 \( 1 - 8.49T + 43T^{2} \)
47 \( 1 - 5.49iT - 47T^{2} \)
53 \( 1 - 0.697T + 53T^{2} \)
59 \( 1 + 13.5iT - 59T^{2} \)
61 \( 1 - 7.31iT - 61T^{2} \)
67 \( 1 - 7.12T + 67T^{2} \)
71 \( 1 + 2.24T + 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 - 4.98T + 79T^{2} \)
83 \( 1 + 7.64T + 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 4.63iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.449479118877362445389624079665, −8.038410778599167476917884257235, −7.41490272939537264691116297687, −7.07450015724659164250110759673, −6.08804330558877163458502475395, −5.35320097828075860479005562176, −4.49442532884685855888556149061, −3.74461789695514151820492786307, −2.52933223105965983371464978217, −1.83071187393256485528709770631, 0.16845304265403479948434338769, 0.890354331027695390721001327574, 2.38862430436918491790399536348, 3.48737131308004401256905931917, 4.42708567327100522186944100794, 5.10347139685119746568330761879, 5.67486838793791349668480611977, 6.71159595752272319620983133051, 7.39067146895451634888881760961, 8.101200278563526221270943110324

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