Properties

Label 2-4840-1.1-c1-0-38
Degree $2$
Conductor $4840$
Sign $1$
Analytic cond. $38.6475$
Root an. cond. $6.21671$
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  + 0.614·3-s + 5-s + 2.24·7-s − 2.62·9-s − 5.19·13-s + 0.614·15-s + 3.22·17-s + 7.19·19-s + 1.38·21-s + 5.11·23-s + 25-s − 3.45·27-s − 9.61·29-s + 0.491·31-s + 2.24·35-s + 0.350·37-s − 3.19·39-s + 4.42·41-s + 10.9·43-s − 2.62·45-s − 1.03·47-s − 1.95·49-s + 1.98·51-s + 3.75·53-s + 4.42·57-s + 12.7·59-s + 6.38·61-s + ⋯
L(s)  = 1  + 0.355·3-s + 0.447·5-s + 0.848·7-s − 0.873·9-s − 1.44·13-s + 0.158·15-s + 0.783·17-s + 1.65·19-s + 0.301·21-s + 1.06·23-s + 0.200·25-s − 0.665·27-s − 1.78·29-s + 0.0882·31-s + 0.379·35-s + 0.0576·37-s − 0.511·39-s + 0.690·41-s + 1.67·43-s − 0.390·45-s − 0.150·47-s − 0.279·49-s + 0.278·51-s + 0.515·53-s + 0.586·57-s + 1.65·59-s + 0.818·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4840\)    =    \(2^{3} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(38.6475\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.502277146\)
\(L(\frac12)\) \(\approx\) \(2.502277146\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good3 \( 1 - 0.614T + 3T^{2} \)
7 \( 1 - 2.24T + 7T^{2} \)
13 \( 1 + 5.19T + 13T^{2} \)
17 \( 1 - 3.22T + 17T^{2} \)
19 \( 1 - 7.19T + 19T^{2} \)
23 \( 1 - 5.11T + 23T^{2} \)
29 \( 1 + 9.61T + 29T^{2} \)
31 \( 1 - 0.491T + 31T^{2} \)
37 \( 1 - 0.350T + 37T^{2} \)
41 \( 1 - 4.42T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + 1.03T + 47T^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 - 6.38T + 61T^{2} \)
67 \( 1 + 8.44T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 + 2.13T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 9.90T + 89T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−8.168654679792975267223237787447, −7.51818985496551099412146756378, −7.16505506423472938895164201659, −5.79403484126939722858293436855, −5.40306800534211026700822669148, −4.76649355437652656550760905108, −3.59525854738450272153750697130, −2.78639681382574888451728858589, −2.05453474023109268457619947334, −0.865543980655330629100108664435, 0.865543980655330629100108664435, 2.05453474023109268457619947334, 2.78639681382574888451728858589, 3.59525854738450272153750697130, 4.76649355437652656550760905108, 5.40306800534211026700822669148, 5.79403484126939722858293436855, 7.16505506423472938895164201659, 7.51818985496551099412146756378, 8.168654679792975267223237787447

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