Properties

Label 2-1045-1.1-c5-0-177
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $167.601$
Root an. cond. $12.9460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4.23·2-s + 30.2·3-s − 14.0·4-s + 25·5-s + 128.·6-s − 150.·7-s − 195.·8-s + 674.·9-s + 105.·10-s + 121·11-s − 425.·12-s + 588.·13-s − 637.·14-s + 757.·15-s − 377.·16-s + 1.11e3·17-s + 2.85e3·18-s + 361·19-s − 351.·20-s − 4.56e3·21-s + 512.·22-s − 3.97e3·23-s − 5.91e3·24-s + 625·25-s + 2.49e3·26-s + 1.30e4·27-s + 2.11e3·28-s + ⋯
L(s)  = 1  + 0.749·2-s + 1.94·3-s − 0.438·4-s + 0.447·5-s + 1.45·6-s − 1.16·7-s − 1.07·8-s + 2.77·9-s + 0.334·10-s + 0.301·11-s − 0.853·12-s + 0.965·13-s − 0.869·14-s + 0.869·15-s − 0.368·16-s + 0.932·17-s + 2.07·18-s + 0.229·19-s − 0.196·20-s − 2.25·21-s + 0.225·22-s − 1.56·23-s − 2.09·24-s + 0.200·25-s + 0.723·26-s + 3.45·27-s + 0.509·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(167.601\)
Root analytic conductor: \(12.9460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(6.733722905\)
\(L(\frac12)\) \(\approx\) \(6.733722905\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 25T \)
11 \( 1 - 121T \)
19 \( 1 - 361T \)
good2 \( 1 - 4.23T + 32T^{2} \)
3 \( 1 - 30.2T + 243T^{2} \)
7 \( 1 + 150.T + 1.68e4T^{2} \)
13 \( 1 - 588.T + 3.71e5T^{2} \)
17 \( 1 - 1.11e3T + 1.41e6T^{2} \)
23 \( 1 + 3.97e3T + 6.43e6T^{2} \)
29 \( 1 + 2.15e3T + 2.05e7T^{2} \)
31 \( 1 - 7.38e3T + 2.86e7T^{2} \)
37 \( 1 + 203.T + 6.93e7T^{2} \)
41 \( 1 - 1.10e3T + 1.15e8T^{2} \)
43 \( 1 - 1.01e4T + 1.47e8T^{2} \)
47 \( 1 - 1.49e4T + 2.29e8T^{2} \)
53 \( 1 + 3.74e4T + 4.18e8T^{2} \)
59 \( 1 - 4.91e4T + 7.14e8T^{2} \)
61 \( 1 - 1.53e4T + 8.44e8T^{2} \)
67 \( 1 + 5.13e4T + 1.35e9T^{2} \)
71 \( 1 + 3.63e4T + 1.80e9T^{2} \)
73 \( 1 - 3.61e4T + 2.07e9T^{2} \)
79 \( 1 - 6.45e4T + 3.07e9T^{2} \)
83 \( 1 - 4.44e4T + 3.93e9T^{2} \)
89 \( 1 - 7.67e4T + 5.58e9T^{2} \)
97 \( 1 - 6.11e4T + 8.58e9T^{2} \)
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

−9.249574130577286745550127674267, −8.465957144878015664207049932787, −7.74041362349792789322258940113, −6.55078677324073310244937784710, −5.85243128680438324200068657972, −4.43676975079173503765666412427, −3.63836435795590937771570244328, −3.20984461398498254590075959500, −2.22544250262652186357866751494, −0.940826447870951414360295884799, 0.940826447870951414360295884799, 2.22544250262652186357866751494, 3.20984461398498254590075959500, 3.63836435795590937771570244328, 4.43676975079173503765666412427, 5.85243128680438324200068657972, 6.55078677324073310244937784710, 7.74041362349792789322258940113, 8.465957144878015664207049932787, 9.249574130577286745550127674267

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