Properties

Label 2-105-105.104-c3-0-13
Degree $2$
Conductor $105$
Sign $0.860 - 0.509i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.64 − 4.47i)3-s − 8·4-s + 11.1i·5-s + 18.5·7-s + (−13.0 + 23.6i)9-s − 11.8i·11-s + (21.1 + 35.7i)12-s + 84.6·13-s + (50.0 − 29.5i)15-s + 64·16-s + 102. i·17-s − 89.4i·20-s + (−49.0 − 82.8i)21-s − 125.·25-s + (140. − 4.47i)27-s − 148.·28-s + ⋯
L(s)  = 1  + (−0.509 − 0.860i)3-s − 4-s + 0.999i·5-s + 0.999·7-s + (−0.481 + 0.876i)9-s − 0.324i·11-s + (0.509 + 0.860i)12-s + 1.80·13-s + (0.860 − 0.509i)15-s + 16-s + 1.46i·17-s − 0.999i·20-s + (−0.509 − 0.860i)21-s − 1.00·25-s + (0.999 − 0.0318i)27-s − 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.860 - 0.509i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.860 - 0.509i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.08372 + 0.296564i\)
\(L(\frac12)\) \(\approx\) \(1.08372 + 0.296564i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.64 + 4.47i)T \)
5 \( 1 - 11.1iT \)
7 \( 1 - 18.5T \)
good2 \( 1 + 8T^{2} \)
11 \( 1 + 11.8iT - 1.33e3T^{2} \)
13 \( 1 - 84.6T + 2.19e3T^{2} \)
17 \( 1 - 102. iT - 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 307. iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 178. iT - 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 863. iT - 3.57e5T^{2} \)
73 \( 1 - 1.13e3T + 3.89e5T^{2} \)
79 \( 1 - 236T + 4.93e5T^{2} \)
83 \( 1 - 1.51e3iT - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + 963.T + 9.12e5T^{2} \)
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

−13.50117854374726516515483662242, −12.42812687668531733905473248784, −11.05849793207044871144049849024, −10.63846150245199012821570316289, −8.675200589701545521645104493429, −7.963437349696721112276010922968, −6.48505104499050042421961108682, −5.43131296013000440657097460426, −3.70366538254294274806929077678, −1.42538772465813365357571217180, 0.824560590213629432363638603045, 4.00620415773054123544866592204, 4.83257541359668209823974145049, 5.81046875853650777734471614348, 8.083948643330592681195056613167, 8.947711014294686640449541922188, 9.792223646675403975162747027153, 11.15539286371327506422488200490, 11.99631013762628393303348693809, 13.33145176651158429888769440093

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