Properties

Label 2-4560-1.1-c1-0-35
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 + 3.30·7-s + 9-s − 2.24·11-s − 3.19·13-s + 15-s + 3.55·17-s + 19-s + 3.30·21-s − 1.55·23-s + 25-s + 27-s + 4.24·29-s + 7.43·31-s − 2.24·33-s + 3.30·35-s − 5.30·37-s − 3.19·39-s + 2.36·41-s − 3.80·43-s + 45-s + 9.55·47-s + 3.94·49-s + 3.55·51-s − 3.55·53-s − 2.24·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.25·7-s + 0.333·9-s − 0.678·11-s − 0.884·13-s + 0.258·15-s + 0.862·17-s + 0.229·19-s + 0.721·21-s − 0.324·23-s + 0.200·25-s + 0.192·27-s + 0.789·29-s + 1.33·31-s − 0.391·33-s + 0.559·35-s − 0.872·37-s − 0.510·39-s + 0.369·41-s − 0.580·43-s + 0.149·45-s + 1.39·47-s + 0.563·49-s + 0.498·51-s − 0.488·53-s − 0.303·55-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

\(L(1)\) \(\approx\) \(3.094690156\)
\(L(\frac12)\) \(\approx\) \(3.094690156\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.30T + 7T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 + 3.19T + 13T^{2} \)
17 \( 1 - 3.55T + 17T^{2} \)
23 \( 1 + 1.55T + 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 7.43T + 31T^{2} \)
37 \( 1 + 5.30T + 37T^{2} \)
41 \( 1 - 2.36T + 41T^{2} \)
43 \( 1 + 3.80T + 43T^{2} \)
47 \( 1 - 9.55T + 47T^{2} \)
53 \( 1 + 3.55T + 53T^{2} \)
59 \( 1 + 0.498T + 59T^{2} \)
61 \( 1 + 1.17T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + 0.117T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 - 7.68T + 97T^{2} \)
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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

−8.117338368412088124412081920818, −7.85847854663916155851515729755, −7.05323430166268022863876290340, −6.11913622713502240646875167457, −5.05972187290822889818492981446, −4.89613470246475703920396556132, −3.73646607515698203792198350918, −2.70171575211544791274023813309, −2.07185647403809080253698421841, −0.998472892298478562306170931813, 0.998472892298478562306170931813, 2.07185647403809080253698421841, 2.70171575211544791274023813309, 3.73646607515698203792198350918, 4.89613470246475703920396556132, 5.05972187290822889818492981446, 6.11913622713502240646875167457, 7.05323430166268022863876290340, 7.85847854663916155851515729755, 8.117338368412088124412081920818

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