Properties

Label 2-2320-1.1-c1-0-32
Degree $2$
Conductor $2320$
Sign $1$
Analytic cond. $18.5252$
Root an. cond. $4.30410$
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  + 2.67·3-s − 5-s + 4.67·7-s + 4.14·9-s + 0.672·11-s − 1.14·13-s − 2.67·15-s + 3.52·17-s − 5.52·19-s + 12.4·21-s + 3.81·23-s + 25-s + 3.05·27-s − 29-s + 1.52·31-s + 1.79·33-s − 4.67·35-s + 7.16·37-s − 3.05·39-s + 2.85·41-s − 8.96·43-s − 4.14·45-s + 6.67·47-s + 14.8·49-s + 9.43·51-s + 10.4·53-s − 0.672·55-s + ⋯
L(s)  = 1  + 1.54·3-s − 0.447·5-s + 1.76·7-s + 1.38·9-s + 0.202·11-s − 0.317·13-s − 0.690·15-s + 0.855·17-s − 1.26·19-s + 2.72·21-s + 0.795·23-s + 0.200·25-s + 0.588·27-s − 0.185·29-s + 0.274·31-s + 0.313·33-s − 0.789·35-s + 1.17·37-s − 0.489·39-s + 0.446·41-s − 1.36·43-s − 0.617·45-s + 0.973·47-s + 2.11·49-s + 1.32·51-s + 1.44·53-s − 0.0907·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(18.5252\)
Root analytic conductor: \(4.30410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2320,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.619925626\)
\(L(\frac12)\) \(\approx\) \(3.619925626\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
29 \( 1 + T \)
good3 \( 1 - 2.67T + 3T^{2} \)
7 \( 1 - 4.67T + 7T^{2} \)
11 \( 1 - 0.672T + 11T^{2} \)
13 \( 1 + 1.14T + 13T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 + 5.52T + 19T^{2} \)
23 \( 1 - 3.81T + 23T^{2} \)
31 \( 1 - 1.52T + 31T^{2} \)
37 \( 1 - 7.16T + 37T^{2} \)
41 \( 1 - 2.85T + 41T^{2} \)
43 \( 1 + 8.96T + 43T^{2} \)
47 \( 1 - 6.67T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 - 7.81T + 67T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + 4.96T + 73T^{2} \)
79 \( 1 + 2.38T + 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 + 1.63T + 89T^{2} \)
97 \( 1 + 9.32T + 97T^{2} \)
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

−8.687680854799441387316241402049, −8.330445210918978505813524479566, −7.65796157152496869288335528901, −7.14482162110104652852548011775, −5.78514484794685701332615633680, −4.65491590974454258454652097099, −4.19656326550927076822799169220, −3.11792587591189350917308135677, −2.22720248327096443703740557520, −1.31277044420626135305584383448, 1.31277044420626135305584383448, 2.22720248327096443703740557520, 3.11792587591189350917308135677, 4.19656326550927076822799169220, 4.65491590974454258454652097099, 5.78514484794685701332615633680, 7.14482162110104652852548011775, 7.65796157152496869288335528901, 8.330445210918978505813524479566, 8.687680854799441387316241402049

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