Properties

Label 2-3330-1.1-c1-0-29
Degree $2$
Conductor $3330$
Sign $1$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 3·7-s + 8-s − 10-s + 5·11-s − 2·13-s + 3·14-s + 16-s + 7·17-s − 2·19-s − 20-s + 5·22-s − 4·23-s + 25-s − 2·26-s + 3·28-s + 5·29-s − 7·31-s + 32-s + 7·34-s − 3·35-s + 37-s − 2·38-s − 40-s − 41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s − 0.316·10-s + 1.50·11-s − 0.554·13-s + 0.801·14-s + 1/4·16-s + 1.69·17-s − 0.458·19-s − 0.223·20-s + 1.06·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.566·28-s + 0.928·29-s − 1.25·31-s + 0.176·32-s + 1.20·34-s − 0.507·35-s + 0.164·37-s − 0.324·38-s − 0.158·40-s − 0.156·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.562837601\)
\(L(\frac12)\) \(\approx\) \(3.562837601\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
37 \( 1 - T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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

−8.479363894691396823198428766432, −7.73463508398268072117254848694, −7.22405775820118106566112790586, −6.24681831042771853874863288742, −5.54105138781531173276253848284, −4.65945553858662436400984934494, −4.05434382945432804672464758693, −3.27535295134737187498488023443, −2.02898702699814251887552526958, −1.11392896224171524842174576504, 1.11392896224171524842174576504, 2.02898702699814251887552526958, 3.27535295134737187498488023443, 4.05434382945432804672464758693, 4.65945553858662436400984934494, 5.54105138781531173276253848284, 6.24681831042771853874863288742, 7.22405775820118106566112790586, 7.73463508398268072117254848694, 8.479363894691396823198428766432

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