Properties

Label 2-4840-1.1-c1-0-13
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  + 0.719·3-s − 5-s − 4.18·7-s − 2.48·9-s + 6.05·13-s − 0.719·15-s − 6.52·17-s − 0.739·19-s − 3.01·21-s − 5.13·23-s + 25-s − 3.94·27-s − 2.08·29-s − 1.12·31-s + 4.18·35-s + 7.11·37-s + 4.35·39-s + 4.02·41-s − 6.29·43-s + 2.48·45-s + 9.24·47-s + 10.5·49-s − 4.69·51-s + 7.77·53-s − 0.532·57-s + 6.16·59-s − 14.6·61-s + ⋯
L(s)  = 1  + 0.415·3-s − 0.447·5-s − 1.58·7-s − 0.827·9-s + 1.67·13-s − 0.185·15-s − 1.58·17-s − 0.169·19-s − 0.657·21-s − 1.06·23-s + 0.200·25-s − 0.759·27-s − 0.386·29-s − 0.202·31-s + 0.707·35-s + 1.16·37-s + 0.697·39-s + 0.628·41-s − 0.960·43-s + 0.370·45-s + 1.34·47-s + 1.50·49-s − 0.657·51-s + 1.06·53-s − 0.0704·57-s + 0.802·59-s − 1.88·61-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

\(L(1)\) \(\approx\) \(1.053579456\)
\(L(\frac12)\) \(\approx\) \(1.053579456\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.719T + 3T^{2} \)
7 \( 1 + 4.18T + 7T^{2} \)
13 \( 1 - 6.05T + 13T^{2} \)
17 \( 1 + 6.52T + 17T^{2} \)
19 \( 1 + 0.739T + 19T^{2} \)
23 \( 1 + 5.13T + 23T^{2} \)
29 \( 1 + 2.08T + 29T^{2} \)
31 \( 1 + 1.12T + 31T^{2} \)
37 \( 1 - 7.11T + 37T^{2} \)
41 \( 1 - 4.02T + 41T^{2} \)
43 \( 1 + 6.29T + 43T^{2} \)
47 \( 1 - 9.24T + 47T^{2} \)
53 \( 1 - 7.77T + 53T^{2} \)
59 \( 1 - 6.16T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + 5.53T + 67T^{2} \)
71 \( 1 - 8.87T + 71T^{2} \)
73 \( 1 - 8.09T + 73T^{2} \)
79 \( 1 - 0.570T + 79T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 + 1.33T + 89T^{2} \)
97 \( 1 - 9.96T + 97T^{2} \)
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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

−8.371776932158123916185947805261, −7.66298794790241295063018158650, −6.60682845102541935126902207688, −6.26524850297985904557315536794, −5.58436406966819962892678321039, −4.18498134873569565370128340674, −3.75172165126452378676412463624, −2.96744685437019887943909861947, −2.13601219011702008617963075010, −0.52245304787648355310800193667, 0.52245304787648355310800193667, 2.13601219011702008617963075010, 2.96744685437019887943909861947, 3.75172165126452378676412463624, 4.18498134873569565370128340674, 5.58436406966819962892678321039, 6.26524850297985904557315536794, 6.60682845102541935126902207688, 7.66298794790241295063018158650, 8.371776932158123916185947805261

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