Properties

Label 2-4560-1.1-c1-0-42
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 + 4.11·7-s + 9-s + 2.11·11-s + 4.80·13-s − 15-s + 4·17-s − 19-s − 4.11·21-s + 0.691·23-s + 25-s − 27-s + 4.11·29-s + 2·31-s − 2.11·33-s + 4.11·35-s + 3.42·37-s − 4.80·39-s + 5.42·41-s + 2.80·43-s + 45-s − 0.691·47-s + 9.91·49-s − 4·51-s − 2.69·53-s + 2.11·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.55·7-s + 0.333·9-s + 0.637·11-s + 1.33·13-s − 0.258·15-s + 0.970·17-s − 0.229·19-s − 0.897·21-s + 0.144·23-s + 0.200·25-s − 0.192·27-s + 0.763·29-s + 0.359·31-s − 0.367·33-s + 0.695·35-s + 0.562·37-s − 0.769·39-s + 0.846·41-s + 0.427·43-s + 0.149·45-s − 0.100·47-s + 1.41·49-s − 0.560·51-s − 0.369·53-s + 0.284·55-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

\(L(1)\) \(\approx\) \(2.674518033\)
\(L(\frac12)\) \(\approx\) \(2.674518033\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.11T + 7T^{2} \)
11 \( 1 - 2.11T + 11T^{2} \)
13 \( 1 - 4.80T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
23 \( 1 - 0.691T + 23T^{2} \)
29 \( 1 - 4.11T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 3.42T + 37T^{2} \)
41 \( 1 - 5.42T + 41T^{2} \)
43 \( 1 - 2.80T + 43T^{2} \)
47 \( 1 + 0.691T + 47T^{2} \)
53 \( 1 + 2.69T + 53T^{2} \)
59 \( 1 + 9.53T + 59T^{2} \)
61 \( 1 + 2.91T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 5.60T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 6.22T + 83T^{2} \)
89 \( 1 + 1.42T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−8.243593692480374962516431046751, −7.73181205847319178499869699827, −6.75751791862380277356311734273, −6.01883073793229236287095203401, −5.49561365059393983937230213453, −4.60362157323689074996289912652, −4.03662244302245061224126887879, −2.84643309068377165297398054818, −1.54321547353715918194906029431, −1.12402033041330793032536183521, 1.12402033041330793032536183521, 1.54321547353715918194906029431, 2.84643309068377165297398054818, 4.03662244302245061224126887879, 4.60362157323689074996289912652, 5.49561365059393983937230213453, 6.01883073793229236287095203401, 6.75751791862380277356311734273, 7.73181205847319178499869699827, 8.243593692480374962516431046751

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