Properties

Label 2-4560-1.1-c1-0-15
Degree $2$
Conductor $4560$
Sign $1$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
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 − 1.41·7-s + 9-s + 0.585·11-s + 0.585·13-s − 15-s − 2.82·17-s − 19-s − 1.41·21-s + 4.82·23-s + 25-s + 27-s − 7.07·29-s + 4.82·31-s + 0.585·33-s + 1.41·35-s + 6.24·37-s + 0.585·39-s + 9.89·41-s − 11.0·43-s − 45-s + 3.17·47-s − 5·49-s − 2.82·51-s + 8.48·53-s − 0.585·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 0.534·7-s + 0.333·9-s + 0.176·11-s + 0.162·13-s − 0.258·15-s − 0.685·17-s − 0.229·19-s − 0.308·21-s + 1.00·23-s + 0.200·25-s + 0.192·27-s − 1.31·29-s + 0.867·31-s + 0.101·33-s + 0.239·35-s + 1.02·37-s + 0.0938·39-s + 1.54·41-s − 1.68·43-s − 0.149·45-s + 0.462·47-s − 0.714·49-s − 0.396·51-s + 1.16·53-s − 0.0789·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.954076480\)
\(L(\frac12)\) \(\approx\) \(1.954076480\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
19 \( 1 + T \)
good7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 - 0.585T + 11T^{2} \)
13 \( 1 - 0.585T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 - 4.82T + 31T^{2} \)
37 \( 1 - 6.24T + 37T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 - 3.17T + 47T^{2} \)
53 \( 1 - 8.48T + 53T^{2} \)
59 \( 1 - 1.17T + 59T^{2} \)
61 \( 1 + 1.65T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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

−8.313621582073231247171322693742, −7.68108958140523198463225719684, −6.87873310912038523048087510399, −6.35428729673210143120297189037, −5.32942091958046848645324615427, −4.42220171777366695328540447564, −3.74318430994777737947033940341, −2.95382513828096291069376149452, −2.07791088339101593293641021023, −0.75048504311458430834985929481, 0.75048504311458430834985929481, 2.07791088339101593293641021023, 2.95382513828096291069376149452, 3.74318430994777737947033940341, 4.42220171777366695328540447564, 5.32942091958046848645324615427, 6.35428729673210143120297189037, 6.87873310912038523048087510399, 7.68108958140523198463225719684, 8.313621582073231247171322693742

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