Properties

Label 2-4560-1.1-c1-0-22
Degree $2$
Conductor $4560$
Sign $1$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 4·7-s + 9-s − 2·11-s + 6·13-s + 15-s − 2·17-s − 19-s − 4·21-s − 6·23-s + 25-s − 27-s + 8·29-s + 8·31-s + 2·33-s − 4·35-s + 10·37-s − 6·39-s − 4·41-s − 4·43-s − 45-s − 6·47-s + 9·49-s + 2·51-s + 12·53-s + 2·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 0.258·15-s − 0.485·17-s − 0.229·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 1.43·31-s + 0.348·33-s − 0.676·35-s + 1.64·37-s − 0.960·39-s − 0.624·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s + 0.280·51-s + 1.64·53-s + 0.269·55-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: yes
Arithmetic: yes
Character: $\chi_{4560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.825826327\)
\(L(\frac12)\) \(\approx\) \(1.825826327\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.356563890293957794000139615142, −7.81438172298263957218751721591, −6.78959424761384834392599078645, −6.11443122339681034548914185781, −5.38303981621335948486439914863, −4.45597351494920758766791387660, −4.16903853251358762455100946022, −2.85853904604941301015742123630, −1.73980526447016001573246886507, −0.821477508523110054438478993132, 0.821477508523110054438478993132, 1.73980526447016001573246886507, 2.85853904604941301015742123630, 4.16903853251358762455100946022, 4.45597351494920758766791387660, 5.38303981621335948486439914863, 6.11443122339681034548914185781, 6.78959424761384834392599078645, 7.81438172298263957218751721591, 8.356563890293957794000139615142

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