Properties

Label 2-1045-1.1-c3-0-103
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  + 5.51·2-s − 6.01·3-s + 22.4·4-s − 5·5-s − 33.1·6-s + 22.7·7-s + 79.8·8-s + 9.17·9-s − 27.5·10-s − 11·11-s − 135.·12-s + 8.83·13-s + 125.·14-s + 30.0·15-s + 261.·16-s + 122.·17-s + 50.6·18-s − 19·19-s − 112.·20-s − 136.·21-s − 60.7·22-s − 177.·23-s − 480.·24-s + 25·25-s + 48.7·26-s + 107.·27-s + 510.·28-s + ⋯
L(s)  = 1  + 1.95·2-s − 1.15·3-s + 2.80·4-s − 0.447·5-s − 2.25·6-s + 1.22·7-s + 3.52·8-s + 0.339·9-s − 0.872·10-s − 0.301·11-s − 3.25·12-s + 0.188·13-s + 2.39·14-s + 0.517·15-s + 4.07·16-s + 1.74·17-s + 0.663·18-s − 0.229·19-s − 1.25·20-s − 1.42·21-s − 0.588·22-s − 1.60·23-s − 4.08·24-s + 0.200·25-s + 0.368·26-s + 0.764·27-s + 3.44·28-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

\(L(2)\) \(\approx\) \(6.015619452\)
\(L(\frac12)\) \(\approx\) \(6.015619452\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.51T + 8T^{2} \)
3 \( 1 + 6.01T + 27T^{2} \)
7 \( 1 - 22.7T + 343T^{2} \)
13 \( 1 - 8.83T + 2.19e3T^{2} \)
17 \( 1 - 122.T + 4.91e3T^{2} \)
23 \( 1 + 177.T + 1.21e4T^{2} \)
29 \( 1 + 36.7T + 2.43e4T^{2} \)
31 \( 1 - 53.3T + 2.97e4T^{2} \)
37 \( 1 - 133.T + 5.06e4T^{2} \)
41 \( 1 + 359.T + 6.89e4T^{2} \)
43 \( 1 - 189.T + 7.95e4T^{2} \)
47 \( 1 - 495.T + 1.03e5T^{2} \)
53 \( 1 - 504.T + 1.48e5T^{2} \)
59 \( 1 - 404.T + 2.05e5T^{2} \)
61 \( 1 - 228.T + 2.26e5T^{2} \)
67 \( 1 - 280.T + 3.00e5T^{2} \)
71 \( 1 - 293.T + 3.57e5T^{2} \)
73 \( 1 + 589.T + 3.89e5T^{2} \)
79 \( 1 - 62.7T + 4.93e5T^{2} \)
83 \( 1 - 856.T + 5.71e5T^{2} \)
89 \( 1 - 923.T + 7.04e5T^{2} \)
97 \( 1 - 340.T + 9.12e5T^{2} \)
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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

−10.25450844785821304651256632540, −8.163745113719053116460065447499, −7.61029044393862697250954741704, −6.59884173899158229597112165623, −5.62133508612956472918838073736, −5.37420924220678844970294295086, −4.42163777369142896752028678834, −3.66459749787791445696512778770, −2.34574059431109824287785504012, −1.10097748345952676146748523543, 1.10097748345952676146748523543, 2.34574059431109824287785504012, 3.66459749787791445696512778770, 4.42163777369142896752028678834, 5.37420924220678844970294295086, 5.62133508612956472918838073736, 6.59884173899158229597112165623, 7.61029044393862697250954741704, 8.163745113719053116460065447499, 10.25450844785821304651256632540

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