Properties

Label 2-1045-1.1-c3-0-121
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.82·2-s + 8.55·3-s + 15.2·4-s − 5·5-s − 41.2·6-s − 30.1·7-s − 35.0·8-s + 46.2·9-s + 24.1·10-s + 11·11-s + 130.·12-s − 23.7·13-s + 145.·14-s − 42.7·15-s + 46.7·16-s − 18.3·17-s − 223.·18-s − 19·19-s − 76.3·20-s − 257.·21-s − 53.0·22-s + 212.·23-s − 299.·24-s + 25·25-s + 114.·26-s + 164.·27-s − 459.·28-s + ⋯
L(s)  = 1  − 1.70·2-s + 1.64·3-s + 1.90·4-s − 0.447·5-s − 2.80·6-s − 1.62·7-s − 1.54·8-s + 1.71·9-s + 0.762·10-s + 0.301·11-s + 3.14·12-s − 0.507·13-s + 2.77·14-s − 0.736·15-s + 0.731·16-s − 0.261·17-s − 2.92·18-s − 0.229·19-s − 0.853·20-s − 2.68·21-s − 0.514·22-s + 1.92·23-s − 2.54·24-s + 0.200·25-s + 0.865·26-s + 1.17·27-s − 3.10·28-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 \( 1 + 4.82T + 8T^{2} \)
3 \( 1 - 8.55T + 27T^{2} \)
7 \( 1 + 30.1T + 343T^{2} \)
13 \( 1 + 23.7T + 2.19e3T^{2} \)
17 \( 1 + 18.3T + 4.91e3T^{2} \)
23 \( 1 - 212.T + 1.21e4T^{2} \)
29 \( 1 - 237.T + 2.43e4T^{2} \)
31 \( 1 + 115.T + 2.97e4T^{2} \)
37 \( 1 - 230.T + 5.06e4T^{2} \)
41 \( 1 + 331.T + 6.89e4T^{2} \)
43 \( 1 - 449.T + 7.95e4T^{2} \)
47 \( 1 - 38.7T + 1.03e5T^{2} \)
53 \( 1 + 345.T + 1.48e5T^{2} \)
59 \( 1 + 888.T + 2.05e5T^{2} \)
61 \( 1 + 818.T + 2.26e5T^{2} \)
67 \( 1 - 701.T + 3.00e5T^{2} \)
71 \( 1 + 193.T + 3.57e5T^{2} \)
73 \( 1 + 1.05e3T + 3.89e5T^{2} \)
79 \( 1 + 1.04e3T + 4.93e5T^{2} \)
83 \( 1 + 443.T + 5.71e5T^{2} \)
89 \( 1 + 938.T + 7.04e5T^{2} \)
97 \( 1 + 1.34e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.083826557201761200836860703199, −8.618297134298029550621965916750, −7.63449698482302903343593455092, −7.09126979965881782438886815444, −6.37270748735187249592203954349, −4.36343978506758938124703677974, −3.03684430708779688394419872091, −2.75587584211636830782526541817, −1.30157788456569080038195051116, 0, 1.30157788456569080038195051116, 2.75587584211636830782526541817, 3.03684430708779688394419872091, 4.36343978506758938124703677974, 6.37270748735187249592203954349, 7.09126979965881782438886815444, 7.63449698482302903343593455092, 8.618297134298029550621965916750, 9.083826557201761200836860703199

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