Properties

Label 2-1045-1.1-c3-0-26
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $61.6569$
Root an. cond. $7.85219$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.620·2-s + 0.467·3-s − 7.61·4-s + 5·5-s + 0.290·6-s − 15.6·7-s − 9.68·8-s − 26.7·9-s + 3.10·10-s + 11·11-s − 3.56·12-s + 20.4·13-s − 9.70·14-s + 2.33·15-s + 54.9·16-s − 120.·17-s − 16.6·18-s − 19·19-s − 38.0·20-s − 7.32·21-s + 6.82·22-s − 40.8·23-s − 4.53·24-s + 25·25-s + 12.6·26-s − 25.1·27-s + 119.·28-s + ⋯
L(s)  = 1  + 0.219·2-s + 0.0900·3-s − 0.951·4-s + 0.447·5-s + 0.0197·6-s − 0.844·7-s − 0.428·8-s − 0.991·9-s + 0.0980·10-s + 0.301·11-s − 0.0857·12-s + 0.435·13-s − 0.185·14-s + 0.0402·15-s + 0.858·16-s − 1.71·17-s − 0.217·18-s − 0.229·19-s − 0.425·20-s − 0.0760·21-s + 0.0661·22-s − 0.370·23-s − 0.0385·24-s + 0.200·25-s + 0.0955·26-s − 0.179·27-s + 0.804·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(61.6569\)
Root analytic conductor: \(7.85219\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.045164708\)
\(L(\frac12)\) \(\approx\) \(1.045164708\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 5T \)
11 \( 1 - 11T \)
19 \( 1 + 19T \)
good2 \( 1 - 0.620T + 8T^{2} \)
3 \( 1 - 0.467T + 27T^{2} \)
7 \( 1 + 15.6T + 343T^{2} \)
13 \( 1 - 20.4T + 2.19e3T^{2} \)
17 \( 1 + 120.T + 4.91e3T^{2} \)
23 \( 1 + 40.8T + 1.21e4T^{2} \)
29 \( 1 - 104.T + 2.43e4T^{2} \)
31 \( 1 + 204.T + 2.97e4T^{2} \)
37 \( 1 - 108.T + 5.06e4T^{2} \)
41 \( 1 + 232.T + 6.89e4T^{2} \)
43 \( 1 - 143.T + 7.95e4T^{2} \)
47 \( 1 - 18.2T + 1.03e5T^{2} \)
53 \( 1 - 579.T + 1.48e5T^{2} \)
59 \( 1 - 446.T + 2.05e5T^{2} \)
61 \( 1 + 83.3T + 2.26e5T^{2} \)
67 \( 1 + 715.T + 3.00e5T^{2} \)
71 \( 1 - 791.T + 3.57e5T^{2} \)
73 \( 1 - 498.T + 3.89e5T^{2} \)
79 \( 1 - 572.T + 4.93e5T^{2} \)
83 \( 1 - 865.T + 5.71e5T^{2} \)
89 \( 1 + 240.T + 7.04e5T^{2} \)
97 \( 1 - 700.T + 9.12e5T^{2} \)
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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

−9.277343797890802548184598471933, −8.927439886006815831965446717525, −8.159567130792477568375351636751, −6.74923366674201065587173358525, −6.09911956241280624191084619837, −5.26045781467760953076037976264, −4.19584361058980836457696395842, −3.34251310391215763920751478071, −2.22734195856135828316217420556, −0.50099925142794268908025464275, 0.50099925142794268908025464275, 2.22734195856135828316217420556, 3.34251310391215763920751478071, 4.19584361058980836457696395842, 5.26045781467760953076037976264, 6.09911956241280624191084619837, 6.74923366674201065587173358525, 8.159567130792477568375351636751, 8.927439886006815831965446717525, 9.277343797890802548184598471933

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