Properties

Label 2-1045-1.1-c3-0-42
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  − 5.27·2-s + 4.83·3-s + 19.8·4-s − 5·5-s − 25.4·6-s − 13.3·7-s − 62.3·8-s − 3.66·9-s + 26.3·10-s + 11·11-s + 95.7·12-s + 44.5·13-s + 70.2·14-s − 24.1·15-s + 170.·16-s + 128.·17-s + 19.3·18-s + 19·19-s − 99.0·20-s − 64.3·21-s − 58.0·22-s + 72.9·23-s − 301.·24-s + 25·25-s − 235.·26-s − 148.·27-s − 263.·28-s + ⋯
L(s)  = 1  − 1.86·2-s + 0.929·3-s + 2.47·4-s − 0.447·5-s − 1.73·6-s − 0.719·7-s − 2.75·8-s − 0.135·9-s + 0.833·10-s + 0.301·11-s + 2.30·12-s + 0.950·13-s + 1.34·14-s − 0.415·15-s + 2.65·16-s + 1.83·17-s + 0.252·18-s + 0.229·19-s − 1.10·20-s − 0.668·21-s − 0.562·22-s + 0.661·23-s − 2.56·24-s + 0.200·25-s − 1.77·26-s − 1.05·27-s − 1.78·28-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

\(L(2)\) \(\approx\) \(0.9810575856\)
\(L(\frac12)\) \(\approx\) \(0.9810575856\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.27T + 8T^{2} \)
3 \( 1 - 4.83T + 27T^{2} \)
7 \( 1 + 13.3T + 343T^{2} \)
13 \( 1 - 44.5T + 2.19e3T^{2} \)
17 \( 1 - 128.T + 4.91e3T^{2} \)
23 \( 1 - 72.9T + 1.21e4T^{2} \)
29 \( 1 + 174.T + 2.43e4T^{2} \)
31 \( 1 + 21.8T + 2.97e4T^{2} \)
37 \( 1 + 65.0T + 5.06e4T^{2} \)
41 \( 1 - 367.T + 6.89e4T^{2} \)
43 \( 1 + 352.T + 7.95e4T^{2} \)
47 \( 1 + 197.T + 1.03e5T^{2} \)
53 \( 1 + 540.T + 1.48e5T^{2} \)
59 \( 1 - 618.T + 2.05e5T^{2} \)
61 \( 1 - 334.T + 2.26e5T^{2} \)
67 \( 1 + 910.T + 3.00e5T^{2} \)
71 \( 1 - 854.T + 3.57e5T^{2} \)
73 \( 1 - 277.T + 3.89e5T^{2} \)
79 \( 1 + 534.T + 4.93e5T^{2} \)
83 \( 1 - 787.T + 5.71e5T^{2} \)
89 \( 1 - 1.14e3T + 7.04e5T^{2} \)
97 \( 1 + 1.16e3T + 9.12e5T^{2} \)
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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

−9.447993931905894334095962190480, −8.746807195842748751396637189784, −8.047267077206532196295934311238, −7.51565230687462457745091651129, −6.57740355828185198555524861967, −5.65070339801017925454648885292, −3.51477499139076041131074112200, −3.10015163713564267290122166512, −1.75002750745644637584349107758, −0.66797565956129901812378318085, 0.66797565956129901812378318085, 1.75002750745644637584349107758, 3.10015163713564267290122166512, 3.51477499139076041131074112200, 5.65070339801017925454648885292, 6.57740355828185198555524861967, 7.51565230687462457745091651129, 8.047267077206532196295934311238, 8.746807195842748751396637189784, 9.447993931905894334095962190480

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