Properties

Label 2-1045-1.1-c3-0-138
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.507·2-s + 7.53·3-s − 7.74·4-s − 5·5-s − 3.82·6-s − 10.8·7-s + 7.98·8-s + 29.8·9-s + 2.53·10-s + 11·11-s − 58.3·12-s − 1.60·13-s + 5.48·14-s − 37.6·15-s + 57.8·16-s + 71.2·17-s − 15.1·18-s − 19·19-s + 38.7·20-s − 81.5·21-s − 5.58·22-s − 83.9·23-s + 60.2·24-s + 25·25-s + 0.813·26-s + 21.4·27-s + 83.7·28-s + ⋯
L(s)  = 1  − 0.179·2-s + 1.45·3-s − 0.967·4-s − 0.447·5-s − 0.260·6-s − 0.584·7-s + 0.353·8-s + 1.10·9-s + 0.0802·10-s + 0.301·11-s − 1.40·12-s − 0.0341·13-s + 0.104·14-s − 0.648·15-s + 0.904·16-s + 1.01·17-s − 0.198·18-s − 0.229·19-s + 0.432·20-s − 0.847·21-s − 0.0540·22-s − 0.761·23-s + 0.512·24-s + 0.200·25-s + 0.00613·26-s + 0.152·27-s + 0.565·28-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: \(1\)
Selberg data: \((2,\ 1045,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.507T + 8T^{2} \)
3 \( 1 - 7.53T + 27T^{2} \)
7 \( 1 + 10.8T + 343T^{2} \)
13 \( 1 + 1.60T + 2.19e3T^{2} \)
17 \( 1 - 71.2T + 4.91e3T^{2} \)
23 \( 1 + 83.9T + 1.21e4T^{2} \)
29 \( 1 + 149.T + 2.43e4T^{2} \)
31 \( 1 - 259.T + 2.97e4T^{2} \)
37 \( 1 - 80.9T + 5.06e4T^{2} \)
41 \( 1 + 440.T + 6.89e4T^{2} \)
43 \( 1 + 138.T + 7.95e4T^{2} \)
47 \( 1 + 29.6T + 1.03e5T^{2} \)
53 \( 1 - 66.4T + 1.48e5T^{2} \)
59 \( 1 - 209.T + 2.05e5T^{2} \)
61 \( 1 - 92.4T + 2.26e5T^{2} \)
67 \( 1 + 110.T + 3.00e5T^{2} \)
71 \( 1 + 522.T + 3.57e5T^{2} \)
73 \( 1 + 525.T + 3.89e5T^{2} \)
79 \( 1 + 897.T + 4.93e5T^{2} \)
83 \( 1 - 895.T + 5.71e5T^{2} \)
89 \( 1 + 1.66e3T + 7.04e5T^{2} \)
97 \( 1 + 420.T + 9.12e5T^{2} \)
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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

−9.063327855423028260182298443158, −8.273722861227687219974318692044, −7.906707930557353134801962713229, −6.85305892905844222184707233444, −5.60395004950110671548361586693, −4.37800653394707491692500549996, −3.64566635955975776387786631806, −2.92443167528093496857004550897, −1.47027909843892168386414759782, 0, 1.47027909843892168386414759782, 2.92443167528093496857004550897, 3.64566635955975776387786631806, 4.37800653394707491692500549996, 5.60395004950110671548361586693, 6.85305892905844222184707233444, 7.906707930557353134801962713229, 8.273722861227687219974318692044, 9.063327855423028260182298443158

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