Properties

Label 2-4840-1.1-c1-0-26
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  − 0.484·3-s + 5-s − 0.484·7-s − 2.76·9-s + 5.28·13-s − 0.484·15-s + 2.48·17-s − 4.73·19-s + 0.235·21-s + 4.24·23-s + 25-s + 2.79·27-s − 3.76·29-s − 0.235·31-s − 0.484·35-s + 5.76·37-s − 2.56·39-s − 0.969·41-s − 0.249·43-s − 2.76·45-s + 3.28·47-s − 6.76·49-s − 1.20·51-s + 5.76·53-s + 2.29·57-s − 12.4·59-s + 9.70·61-s + ⋯
L(s)  = 1  − 0.279·3-s + 0.447·5-s − 0.183·7-s − 0.921·9-s + 1.46·13-s − 0.125·15-s + 0.602·17-s − 1.08·19-s + 0.0513·21-s + 0.886·23-s + 0.200·25-s + 0.537·27-s − 0.699·29-s − 0.0422·31-s − 0.0819·35-s + 0.947·37-s − 0.409·39-s − 0.151·41-s − 0.0380·43-s − 0.412·45-s + 0.478·47-s − 0.966·49-s − 0.168·51-s + 0.791·53-s + 0.304·57-s − 1.62·59-s + 1.24·61-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

\(L(1)\) \(\approx\) \(1.755512153\)
\(L(\frac12)\) \(\approx\) \(1.755512153\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.484T + 3T^{2} \)
7 \( 1 + 0.484T + 7T^{2} \)
13 \( 1 - 5.28T + 13T^{2} \)
17 \( 1 - 2.48T + 17T^{2} \)
19 \( 1 + 4.73T + 19T^{2} \)
23 \( 1 - 4.24T + 23T^{2} \)
29 \( 1 + 3.76T + 29T^{2} \)
31 \( 1 + 0.235T + 31T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 + 0.969T + 41T^{2} \)
43 \( 1 + 0.249T + 43T^{2} \)
47 \( 1 - 3.28T + 47T^{2} \)
53 \( 1 - 5.76T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 9.70T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 + 5.70T + 71T^{2} \)
73 \( 1 + 1.75T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 16.7T + 83T^{2} \)
89 \( 1 - 3.82T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−8.383102869943295827456188439267, −7.60945750707165354965364451305, −6.56893653642224372216609010411, −6.09856857895543836720010814584, −5.53555113740852080194256908235, −4.64505150065000630832444124689, −3.64711951760901079963921666733, −2.94267794980687641536668160063, −1.87584195840494305369279306852, −0.74558954190318923600909187921, 0.74558954190318923600909187921, 1.87584195840494305369279306852, 2.94267794980687641536668160063, 3.64711951760901079963921666733, 4.64505150065000630832444124689, 5.53555113740852080194256908235, 6.09856857895543836720010814584, 6.56893653642224372216609010411, 7.60945750707165354965364451305, 8.383102869943295827456188439267

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