Properties

Label 2-4560-1.1-c1-0-47
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 + 2.52·7-s + 9-s + 3.10·11-s + 6.72·13-s + 15-s − 2.57·17-s + 19-s + 2.52·21-s + 4.57·23-s + 25-s + 27-s − 1.10·29-s − 7.83·31-s + 3.10·33-s + 2.52·35-s − 4.52·37-s + 6.72·39-s + 6.15·41-s + 7.68·43-s + 45-s + 3.42·47-s − 0.627·49-s − 2.57·51-s + 2.57·53-s + 3.10·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.954·7-s + 0.333·9-s + 0.935·11-s + 1.86·13-s + 0.258·15-s − 0.625·17-s + 0.229·19-s + 0.550·21-s + 0.954·23-s + 0.200·25-s + 0.192·27-s − 0.204·29-s − 1.40·31-s + 0.540·33-s + 0.426·35-s − 0.743·37-s + 1.07·39-s + 0.960·41-s + 1.17·43-s + 0.149·45-s + 0.499·47-s − 0.0896·49-s − 0.361·51-s + 0.354·53-s + 0.418·55-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

\(L(1)\) \(\approx\) \(3.642977804\)
\(L(\frac12)\) \(\approx\) \(3.642977804\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.52T + 7T^{2} \)
11 \( 1 - 3.10T + 11T^{2} \)
13 \( 1 - 6.72T + 13T^{2} \)
17 \( 1 + 2.57T + 17T^{2} \)
23 \( 1 - 4.57T + 23T^{2} \)
29 \( 1 + 1.10T + 29T^{2} \)
31 \( 1 + 7.83T + 31T^{2} \)
37 \( 1 + 4.52T + 37T^{2} \)
41 \( 1 - 6.15T + 41T^{2} \)
43 \( 1 - 7.68T + 43T^{2} \)
47 \( 1 - 3.42T + 47T^{2} \)
53 \( 1 - 2.57T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + 8.41T + 67T^{2} \)
71 \( 1 + 9.45T + 71T^{2} \)
73 \( 1 + 9.25T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4.47T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−8.612146907409701480637514716344, −7.60117422152673220125750841729, −6.97661411185707920819498895304, −6.09673648229398328550240466067, −5.48611932564853712236000164268, −4.40479018708109083822113919629, −3.85076342794092243991717441785, −2.90736949869887144200165221752, −1.73957985921372683655496556478, −1.20998597371058726737610643989, 1.20998597371058726737610643989, 1.73957985921372683655496556478, 2.90736949869887144200165221752, 3.85076342794092243991717441785, 4.40479018708109083822113919629, 5.48611932564853712236000164268, 6.09673648229398328550240466067, 6.97661411185707920819498895304, 7.60117422152673220125750841729, 8.612146907409701480637514716344

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