Properties

Label 2-4560-1.1-c1-0-7
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 − 0.911·7-s + 9-s − 2.91·11-s − 6.25·13-s − 15-s + 4·17-s − 19-s + 0.911·21-s − 5.34·23-s + 25-s − 27-s − 0.911·29-s + 2·31-s + 2.91·33-s − 0.911·35-s + 4.43·37-s + 6.25·39-s + 6.43·41-s − 8.25·43-s + 45-s + 5.34·47-s − 6.16·49-s − 4·51-s + 3.34·53-s − 2.91·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.344·7-s + 0.333·9-s − 0.877·11-s − 1.73·13-s − 0.258·15-s + 0.970·17-s − 0.229·19-s + 0.198·21-s − 1.11·23-s + 0.200·25-s − 0.192·27-s − 0.169·29-s + 0.359·31-s + 0.506·33-s − 0.154·35-s + 0.729·37-s + 1.00·39-s + 1.00·41-s − 1.25·43-s + 0.149·45-s + 0.779·47-s − 0.881·49-s − 0.560·51-s + 0.459·53-s − 0.392·55-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

\(L(1)\) \(\approx\) \(1.083966730\)
\(L(\frac12)\) \(\approx\) \(1.083966730\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.911T + 7T^{2} \)
11 \( 1 + 2.91T + 11T^{2} \)
13 \( 1 + 6.25T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
23 \( 1 + 5.34T + 23T^{2} \)
29 \( 1 + 0.911T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 4.43T + 37T^{2} \)
41 \( 1 - 6.43T + 41T^{2} \)
43 \( 1 + 8.25T + 43T^{2} \)
47 \( 1 - 5.34T + 47T^{2} \)
53 \( 1 - 3.34T + 53T^{2} \)
59 \( 1 + 5.52T + 59T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 16.5T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 3.82T + 83T^{2} \)
89 \( 1 + 2.43T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−8.048988434201979413023615469983, −7.68223062242674555336878177313, −6.80629329401620300053390826437, −6.09980711429328337916898057058, −5.32183268793334977547399771920, −4.87250610194109816541097932883, −3.83358719649212264276649746141, −2.73392550128240915909749650632, −2.04416368777916527683001091977, −0.57158648188473683718577188158, 0.57158648188473683718577188158, 2.04416368777916527683001091977, 2.73392550128240915909749650632, 3.83358719649212264276649746141, 4.87250610194109816541097932883, 5.32183268793334977547399771920, 6.09980711429328337916898057058, 6.80629329401620300053390826437, 7.68223062242674555336878177313, 8.048988434201979413023615469983

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