Properties

Label 2-3040-1.1-c1-0-19
Degree $2$
Conductor $3040$
Sign $1$
Analytic cond. $24.2745$
Root an. cond. $4.92691$
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  − 0.428·3-s − 5-s + 4.67·7-s − 2.81·9-s + 2.67·11-s − 2.24·13-s + 0.428·15-s + 2·17-s + 19-s − 2·21-s − 0.672·23-s + 25-s + 2.48·27-s + 1.14·29-s + 1.14·31-s − 1.14·33-s − 4.67·35-s + 4.24·37-s + 0.960·39-s + 4.85·41-s − 5.52·43-s + 2.81·45-s + 4.67·47-s + 14.8·49-s − 0.856·51-s − 0.244·53-s − 2.67·55-s + ⋯
L(s)  = 1  − 0.247·3-s − 0.447·5-s + 1.76·7-s − 0.938·9-s + 0.805·11-s − 0.622·13-s + 0.110·15-s + 0.485·17-s + 0.229·19-s − 0.436·21-s − 0.140·23-s + 0.200·25-s + 0.479·27-s + 0.212·29-s + 0.205·31-s − 0.199·33-s − 0.789·35-s + 0.697·37-s + 0.153·39-s + 0.758·41-s − 0.843·43-s + 0.419·45-s + 0.681·47-s + 2.11·49-s − 0.119·51-s − 0.0336·53-s − 0.360·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(24.2745\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.846325674\)
\(L(\frac12)\) \(\approx\) \(1.846325674\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 + 0.428T + 3T^{2} \)
7 \( 1 - 4.67T + 7T^{2} \)
11 \( 1 - 2.67T + 11T^{2} \)
13 \( 1 + 2.24T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
23 \( 1 + 0.672T + 23T^{2} \)
29 \( 1 - 1.14T + 29T^{2} \)
31 \( 1 - 1.14T + 31T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 - 4.85T + 41T^{2} \)
43 \( 1 + 5.52T + 43T^{2} \)
47 \( 1 - 4.67T + 47T^{2} \)
53 \( 1 + 0.244T + 53T^{2} \)
59 \( 1 - 0.287T + 59T^{2} \)
61 \( 1 + 8.87T + 61T^{2} \)
67 \( 1 + 9.20T + 67T^{2} \)
71 \( 1 - 9.14T + 71T^{2} \)
73 \( 1 - 1.14T + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 0.183T + 83T^{2} \)
89 \( 1 - 9.14T + 89T^{2} \)
97 \( 1 - 9.10T + 97T^{2} \)
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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

−8.638239429612075943410627899427, −7.902884238663875921221679985740, −7.47380689408166588316036958665, −6.39328922219487964270446671202, −5.54903913786008379698450055072, −4.84481773471542965328947219505, −4.19325285995919837500300503544, −3.06802362097335032790066460448, −1.97747545382210122743039473936, −0.867849798057032037100840160828, 0.867849798057032037100840160828, 1.97747545382210122743039473936, 3.06802362097335032790066460448, 4.19325285995919837500300503544, 4.84481773471542965328947219505, 5.54903913786008379698450055072, 6.39328922219487964270446671202, 7.47380689408166588316036958665, 7.902884238663875921221679985740, 8.638239429612075943410627899427

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