Properties

Label 2-4560-1.1-c1-0-33
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 3.41·7-s + 9-s + 1.41·11-s + 4.24·13-s + 15-s + 2.82·17-s + 19-s − 3.41·21-s + 4.82·23-s + 25-s − 27-s + 2.24·29-s + 8.82·31-s − 1.41·33-s − 3.41·35-s − 7.07·37-s − 4.24·39-s + 2.24·41-s + 1.75·43-s − 45-s − 4.82·47-s + 4.65·49-s − 2.82·51-s − 12.4·53-s − 1.41·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.29·7-s + 0.333·9-s + 0.426·11-s + 1.17·13-s + 0.258·15-s + 0.685·17-s + 0.229·19-s − 0.745·21-s + 1.00·23-s + 0.200·25-s − 0.192·27-s + 0.416·29-s + 1.58·31-s − 0.246·33-s − 0.577·35-s − 1.16·37-s − 0.679·39-s + 0.350·41-s + 0.267·43-s − 0.149·45-s − 0.704·47-s + 0.665·49-s − 0.396·51-s − 1.71·53-s − 0.190·55-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

\(L(1)\) \(\approx\) \(2.093898191\)
\(L(\frac12)\) \(\approx\) \(2.093898191\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 - 2.24T + 29T^{2} \)
31 \( 1 - 8.82T + 31T^{2} \)
37 \( 1 + 7.07T + 37T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 - 1.75T + 43T^{2} \)
47 \( 1 + 4.82T + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 5.17T + 71T^{2} \)
73 \( 1 + 3.65T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.232734835395951866786952815297, −7.73372965530860845693000267229, −6.80101487378957457616175655600, −6.19425871892741556483277878447, −5.23025417682103031654133567100, −4.73638373518931480150586112572, −3.89446926439897743578843011575, −3.01785667320687823454346050223, −1.58748057196747311901682976967, −0.937794634178985329137266693806, 0.937794634178985329137266693806, 1.58748057196747311901682976967, 3.01785667320687823454346050223, 3.89446926439897743578843011575, 4.73638373518931480150586112572, 5.23025417682103031654133567100, 6.19425871892741556483277878447, 6.80101487378957457616175655600, 7.73372965530860845693000267229, 8.232734835395951866786952815297

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