Properties

Label 2-1045-1.1-c3-0-141
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.96·2-s − 9.59·3-s + 16.6·4-s − 5·5-s − 47.6·6-s − 3.47·7-s + 42.8·8-s + 65.0·9-s − 24.8·10-s − 11·11-s − 159.·12-s + 11.2·13-s − 17.2·14-s + 47.9·15-s + 79.4·16-s − 2.14·17-s + 322.·18-s + 19·19-s − 83.1·20-s + 33.3·21-s − 54.5·22-s + 117.·23-s − 410.·24-s + 25·25-s + 55.7·26-s − 365.·27-s − 57.7·28-s + ⋯
L(s)  = 1  + 1.75·2-s − 1.84·3-s + 2.07·4-s − 0.447·5-s − 3.23·6-s − 0.187·7-s + 1.89·8-s + 2.40·9-s − 0.784·10-s − 0.301·11-s − 3.83·12-s + 0.239·13-s − 0.328·14-s + 0.825·15-s + 1.24·16-s − 0.0306·17-s + 4.22·18-s + 0.229·19-s − 0.929·20-s + 0.346·21-s − 0.529·22-s + 1.06·23-s − 3.49·24-s + 0.200·25-s + 0.420·26-s − 2.60·27-s − 0.389·28-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: $\chi_{1045} (1, \cdot )$
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 \( 1 - 4.96T + 8T^{2} \)
3 \( 1 + 9.59T + 27T^{2} \)
7 \( 1 + 3.47T + 343T^{2} \)
13 \( 1 - 11.2T + 2.19e3T^{2} \)
17 \( 1 + 2.14T + 4.91e3T^{2} \)
23 \( 1 - 117.T + 1.21e4T^{2} \)
29 \( 1 + 57.3T + 2.43e4T^{2} \)
31 \( 1 + 262.T + 2.97e4T^{2} \)
37 \( 1 - 311.T + 5.06e4T^{2} \)
41 \( 1 - 195.T + 6.89e4T^{2} \)
43 \( 1 + 431.T + 7.95e4T^{2} \)
47 \( 1 + 276.T + 1.03e5T^{2} \)
53 \( 1 + 39.9T + 1.48e5T^{2} \)
59 \( 1 + 618.T + 2.05e5T^{2} \)
61 \( 1 + 44.3T + 2.26e5T^{2} \)
67 \( 1 + 770.T + 3.00e5T^{2} \)
71 \( 1 - 843.T + 3.57e5T^{2} \)
73 \( 1 + 475.T + 3.89e5T^{2} \)
79 \( 1 + 605.T + 4.93e5T^{2} \)
83 \( 1 - 118.T + 5.71e5T^{2} \)
89 \( 1 + 641.T + 7.04e5T^{2} \)
97 \( 1 - 550.T + 9.12e5T^{2} \)
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

−9.427601720884995921375714944265, −7.70050525614920174916614129758, −6.92318368981481894045947625767, −6.26299067588290300039334897880, −5.50542190799375244070698141992, −4.89234144636434192099953867418, −4.14211835857289026086553504781, −3.11959324979876191163899397296, −1.49666904229627869351072453029, 0, 1.49666904229627869351072453029, 3.11959324979876191163899397296, 4.14211835857289026086553504781, 4.89234144636434192099953867418, 5.50542190799375244070698141992, 6.26299067588290300039334897880, 6.92318368981481894045947625767, 7.70050525614920174916614129758, 9.427601720884995921375714944265

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