Properties

Label 2-4840-1.1-c1-0-77
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  + 3.35·3-s − 5-s + 3.81·7-s + 8.24·9-s − 0.459·13-s − 3.35·15-s + 7.78·17-s − 1.54·19-s + 12.7·21-s − 8.24·23-s + 25-s + 17.6·27-s + 6.70·29-s − 7.54·31-s − 3.81·35-s + 4.70·37-s − 1.54·39-s + 1.45·41-s − 3.97·43-s − 8.24·45-s + 0.272·47-s + 7.54·49-s + 26.1·51-s − 2.62·53-s − 5.16·57-s − 8.24·59-s − 0.751·61-s + ⋯
L(s)  = 1  + 1.93·3-s − 0.447·5-s + 1.44·7-s + 2.74·9-s − 0.127·13-s − 0.865·15-s + 1.88·17-s − 0.353·19-s + 2.79·21-s − 1.71·23-s + 0.200·25-s + 3.38·27-s + 1.24·29-s − 1.35·31-s − 0.644·35-s + 0.773·37-s − 0.246·39-s + 0.227·41-s − 0.606·43-s − 1.22·45-s + 0.0397·47-s + 1.07·49-s + 3.65·51-s − 0.360·53-s − 0.684·57-s − 1.07·59-s − 0.0962·61-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

\(L(1)\) \(\approx\) \(4.921820322\)
\(L(\frac12)\) \(\approx\) \(4.921820322\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.35T + 3T^{2} \)
7 \( 1 - 3.81T + 7T^{2} \)
13 \( 1 + 0.459T + 13T^{2} \)
17 \( 1 - 7.78T + 17T^{2} \)
19 \( 1 + 1.54T + 19T^{2} \)
23 \( 1 + 8.24T + 23T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 + 7.54T + 31T^{2} \)
37 \( 1 - 4.70T + 37T^{2} \)
41 \( 1 - 1.45T + 41T^{2} \)
43 \( 1 + 3.97T + 43T^{2} \)
47 \( 1 - 0.272T + 47T^{2} \)
53 \( 1 + 2.62T + 53T^{2} \)
59 \( 1 + 8.24T + 59T^{2} \)
61 \( 1 + 0.751T + 61T^{2} \)
67 \( 1 + 4.06T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 0.918T + 73T^{2} \)
79 \( 1 + 5.78T + 79T^{2} \)
83 \( 1 - 9.16T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 0.459T + 97T^{2} \)
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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

−8.177282182851777001880471849156, −7.70604180094609390799688729180, −7.44291272105296379838322533753, −6.18967962981510895347553481301, −5.05171341336327228665907751247, −4.32949982156817093707773293154, −3.68552492525260178220883133372, −2.90029850512863604669258778071, −1.96121844692243863885001563848, −1.28565817090428089194125995173, 1.28565817090428089194125995173, 1.96121844692243863885001563848, 2.90029850512863604669258778071, 3.68552492525260178220883133372, 4.32949982156817093707773293154, 5.05171341336327228665907751247, 6.18967962981510895347553481301, 7.44291272105296379838322533753, 7.70604180094609390799688729180, 8.177282182851777001880471849156

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