Properties

Label 2-1045-1.1-c3-0-66
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.94·2-s − 3.08·3-s + 0.644·4-s + 5·5-s − 9.08·6-s + 21.8·7-s − 21.6·8-s − 17.4·9-s + 14.7·10-s + 11·11-s − 1.99·12-s + 82.8·13-s + 64.1·14-s − 15.4·15-s − 68.7·16-s + 12.6·17-s − 51.3·18-s − 19·19-s + 3.22·20-s − 67.4·21-s + 32.3·22-s − 184.·23-s + 66.8·24-s + 25·25-s + 243.·26-s + 137.·27-s + 14.0·28-s + ⋯
L(s)  = 1  + 1.03·2-s − 0.594·3-s + 0.0805·4-s + 0.447·5-s − 0.618·6-s + 1.17·7-s − 0.955·8-s − 0.646·9-s + 0.464·10-s + 0.301·11-s − 0.0479·12-s + 1.76·13-s + 1.22·14-s − 0.265·15-s − 1.07·16-s + 0.180·17-s − 0.671·18-s − 0.229·19-s + 0.0360·20-s − 0.700·21-s + 0.313·22-s − 1.66·23-s + 0.568·24-s + 0.200·25-s + 1.83·26-s + 0.979·27-s + 0.0949·28-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

\(L(2)\) \(\approx\) \(3.186812219\)
\(L(\frac12)\) \(\approx\) \(3.186812219\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.94T + 8T^{2} \)
3 \( 1 + 3.08T + 27T^{2} \)
7 \( 1 - 21.8T + 343T^{2} \)
13 \( 1 - 82.8T + 2.19e3T^{2} \)
17 \( 1 - 12.6T + 4.91e3T^{2} \)
23 \( 1 + 184.T + 1.21e4T^{2} \)
29 \( 1 - 107.T + 2.43e4T^{2} \)
31 \( 1 + 12.9T + 2.97e4T^{2} \)
37 \( 1 + 57.5T + 5.06e4T^{2} \)
41 \( 1 - 146.T + 6.89e4T^{2} \)
43 \( 1 - 304.T + 7.95e4T^{2} \)
47 \( 1 - 480.T + 1.03e5T^{2} \)
53 \( 1 + 581.T + 1.48e5T^{2} \)
59 \( 1 + 420.T + 2.05e5T^{2} \)
61 \( 1 - 357.T + 2.26e5T^{2} \)
67 \( 1 - 584.T + 3.00e5T^{2} \)
71 \( 1 - 1.17e3T + 3.57e5T^{2} \)
73 \( 1 - 333.T + 3.89e5T^{2} \)
79 \( 1 - 786.T + 4.93e5T^{2} \)
83 \( 1 - 87.8T + 5.71e5T^{2} \)
89 \( 1 - 1.45e3T + 7.04e5T^{2} \)
97 \( 1 + 1.38e3T + 9.12e5T^{2} \)
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

−9.496431189611638169176227558437, −8.593593995515546013560150047702, −8.040249755561486568046256570245, −6.42642510005325526840501326766, −5.97263589617596147119559165159, −5.27436486399365145320812776399, −4.36911497516141879832110286537, −3.54697916532804150617865558059, −2.18305808288351369047894609918, −0.857325965831403738730171627123, 0.857325965831403738730171627123, 2.18305808288351369047894609918, 3.54697916532804150617865558059, 4.36911497516141879832110286537, 5.27436486399365145320812776399, 5.97263589617596147119559165159, 6.42642510005325526840501326766, 8.040249755561486568046256570245, 8.593593995515546013560150047702, 9.496431189611638169176227558437

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