Properties

Label 2-3360-1.1-c1-0-18
Degree $2$
Conductor $3360$
Sign $1$
Analytic cond. $26.8297$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s − 7-s + 9-s + 5.46·11-s + 3.46·13-s − 15-s + 2·17-s + 5.46·19-s − 21-s − 6.92·23-s + 25-s + 27-s − 2·29-s − 5.46·31-s + 5.46·33-s + 35-s + 2·37-s + 3.46·39-s − 4.92·41-s + 4·43-s − 45-s + 10.9·47-s + 49-s + 2·51-s − 0.535·53-s − 5.46·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s + 1.64·11-s + 0.960·13-s − 0.258·15-s + 0.485·17-s + 1.25·19-s − 0.218·21-s − 1.44·23-s + 0.200·25-s + 0.192·27-s − 0.371·29-s − 0.981·31-s + 0.951·33-s + 0.169·35-s + 0.328·37-s + 0.554·39-s − 0.769·41-s + 0.609·43-s − 0.149·45-s + 1.59·47-s + 0.142·49-s + 0.280·51-s − 0.0736·53-s − 0.736·55-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.456887694\)
\(L(\frac12)\) \(\approx\) \(2.456887694\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
good11 \( 1 - 5.46T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 5.46T + 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 4.92T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 0.535T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 + 4.92T + 61T^{2} \)
67 \( 1 + 6.92T + 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 - 0.535T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 - 19.4T + 97T^{2} \)
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

−8.769265212282106275604676518514, −7.78755421768061751153489709550, −7.32061362718310246475164860877, −6.33235717882000149368170322322, −5.80751418458941375949304291698, −4.54899778570738069265596064240, −3.60597315295901231959789155980, −3.45950003571848085079633022356, −1.95995435734588761818177844585, −0.965612699922309252692172287047, 0.965612699922309252692172287047, 1.95995435734588761818177844585, 3.45950003571848085079633022356, 3.60597315295901231959789155980, 4.54899778570738069265596064240, 5.80751418458941375949304291698, 6.33235717882000149368170322322, 7.32061362718310246475164860877, 7.78755421768061751153489709550, 8.769265212282106275604676518514

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