Properties

Label 2-3360-1.1-c1-0-44
Degree $2$
Conductor $3360$
Sign $-1$
Analytic cond. $26.8297$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 7-s + 9-s − 4·11-s − 2·13-s + 15-s − 2·17-s − 21-s + 25-s + 27-s + 6·29-s − 8·31-s − 4·33-s − 35-s + 2·37-s − 2·39-s − 6·41-s − 4·43-s + 45-s + 49-s − 2·51-s − 6·53-s − 4·55-s − 8·59-s − 2·61-s − 63-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s − 0.485·17-s − 0.218·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.169·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1/7·49-s − 0.280·51-s − 0.824·53-s − 0.539·55-s − 1.04·59-s − 0.256·61-s − 0.125·63-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: yes
Arithmetic: yes
Character: $\chi_{3360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3360,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−8.289925822736672182905546770484, −7.52341426380507923940111350048, −6.86982879879750179525552680071, −5.98864617661575263357251261827, −5.14478606985409037175948105142, −4.43780046414078566409971573458, −3.25044325150774504833750932654, −2.62810229682407964863325346977, −1.69090786229433489075097898038, 0, 1.69090786229433489075097898038, 2.62810229682407964863325346977, 3.25044325150774504833750932654, 4.43780046414078566409971573458, 5.14478606985409037175948105142, 5.98864617661575263357251261827, 6.86982879879750179525552680071, 7.52341426380507923940111350048, 8.289925822736672182905546770484

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