Properties

Label 2-4592-1.1-c1-0-28
Degree $2$
Conductor $4592$
Sign $1$
Analytic cond. $36.6673$
Root an. cond. $6.05535$
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  − 0.539·3-s + 4.10·5-s − 7-s − 2.70·9-s − 2.76·11-s − 4.05·13-s − 2.21·15-s + 5.22·17-s + 0.109·19-s + 0.539·21-s − 6.08·23-s + 11.8·25-s + 3.08·27-s + 2.25·29-s + 1.18·31-s + 1.49·33-s − 4.10·35-s − 8.95·37-s + 2.18·39-s − 41-s + 7.93·43-s − 11.1·45-s + 10.5·47-s + 49-s − 2.81·51-s + 6.23·53-s − 11.3·55-s + ⋯
L(s)  = 1  − 0.311·3-s + 1.83·5-s − 0.377·7-s − 0.902·9-s − 0.834·11-s − 1.12·13-s − 0.571·15-s + 1.26·17-s + 0.0250·19-s + 0.117·21-s − 1.26·23-s + 2.36·25-s + 0.592·27-s + 0.418·29-s + 0.212·31-s + 0.260·33-s − 0.693·35-s − 1.47·37-s + 0.350·39-s − 0.156·41-s + 1.21·43-s − 1.65·45-s + 1.54·47-s + 0.142·49-s − 0.394·51-s + 0.855·53-s − 1.53·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4592\)    =    \(2^{4} \cdot 7 \cdot 41\)
Sign: $1$
Analytic conductor: \(36.6673\)
Root analytic conductor: \(6.05535\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4592,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.917841580\)
\(L(\frac12)\) \(\approx\) \(1.917841580\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
41 \( 1 + T \)
good3 \( 1 + 0.539T + 3T^{2} \)
5 \( 1 - 4.10T + 5T^{2} \)
11 \( 1 + 2.76T + 11T^{2} \)
13 \( 1 + 4.05T + 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 - 0.109T + 19T^{2} \)
23 \( 1 + 6.08T + 23T^{2} \)
29 \( 1 - 2.25T + 29T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 + 8.95T + 37T^{2} \)
43 \( 1 - 7.93T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 - 6.23T + 53T^{2} \)
59 \( 1 - 9.43T + 59T^{2} \)
61 \( 1 - 6.89T + 61T^{2} \)
67 \( 1 - 1.35T + 67T^{2} \)
71 \( 1 - 7.90T + 71T^{2} \)
73 \( 1 - 9.88T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + 0.852T + 89T^{2} \)
97 \( 1 - 9.92T + 97T^{2} \)
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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

−8.367053808429108865545553902196, −7.57161015582169034402493197504, −6.66814269352914409146910915293, −6.00900547159851103059615463457, −5.30785121004310912842688439656, −5.16871614348470341787158180926, −3.63988028853190716488082208153, −2.52356896758799149308776611265, −2.24645461374067055315628909247, −0.75819213595598159214198956946, 0.75819213595598159214198956946, 2.24645461374067055315628909247, 2.52356896758799149308776611265, 3.63988028853190716488082208153, 5.16871614348470341787158180926, 5.30785121004310912842688439656, 6.00900547159851103059615463457, 6.66814269352914409146910915293, 7.57161015582169034402493197504, 8.367053808429108865545553902196

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