Properties

Label 2-4560-1.1-c1-0-5
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 + 0.585·7-s + 9-s − 1.41·11-s − 4.24·13-s + 15-s − 2.82·17-s + 19-s − 0.585·21-s − 0.828·23-s + 25-s − 27-s − 6.24·29-s + 3.17·31-s + 1.41·33-s − 0.585·35-s + 7.07·37-s + 4.24·39-s − 6.24·41-s + 10.2·43-s − 45-s + 0.828·47-s − 6.65·49-s + 2.82·51-s + 4.48·53-s + 1.41·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.221·7-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 − 0.127·21-s − 0.172·23-s + 0.200·25-s − 0.192·27-s − 1.15·29-s + 0.569·31-s + 0.246·33-s − 0.0990·35-s + 1.16·37-s + 0.679·39-s − 0.974·41-s + 1.56·43-s − 0.149·45-s + 0.120·47-s − 0.950·49-s + 0.396·51-s + 0.616·53-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

\(L(1)\) \(\approx\) \(0.9349768575\)
\(L(\frac12)\) \(\approx\) \(0.9349768575\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
23 \( 1 + 0.828T + 23T^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
31 \( 1 - 3.17T + 31T^{2} \)
37 \( 1 - 7.07T + 37T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 - 0.828T + 47T^{2} \)
53 \( 1 - 4.48T + 53T^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 - 7.65T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
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

−8.051770561901685873738901575430, −7.66953936516252925853522840146, −6.91076218640348467049664668994, −6.15199153696383977971176696767, −5.27235523143529734396562364263, −4.69661079306401521636875259251, −3.95329200373950509739396774569, −2.83864412546053812854048244951, −1.94355958571798899954313066482, −0.53727142922983314127949709546, 0.53727142922983314127949709546, 1.94355958571798899954313066482, 2.83864412546053812854048244951, 3.95329200373950509739396774569, 4.69661079306401521636875259251, 5.27235523143529734396562364263, 6.15199153696383977971176696767, 6.91076218640348467049664668994, 7.66953936516252925853522840146, 8.051770561901685873738901575430

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