Properties

Label 2-105-105.104-c3-0-1
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

\(L(2)\) \(\approx\) \(0.0412219 + 0.150636i\)
\(L(\frac12)\) \(\approx\) \(0.0412219 + 0.150636i\)
\(L(\frac{5}{2})\) 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

−13.64721458482573942972138324449, −12.81776850780024519160925558774, −11.99978961268032944395236762502, −10.15869819334030884118839516862, −9.533737129037703884664802627089, −8.115709256405823213106949219484, −7.10250025143805530197075874098, −5.95013541487232629109365514668, −3.91598681626432828455324155804, −2.51226379568026950090055357181, 0.079474028909183583181871582819, 3.09800503779469979558425592120, 4.58951868927136206353080724687, 5.32900118313901463602575859670, 7.52709592562571170017705975291, 8.941373085706403648205544576245, 9.391392839877074209176750872784, 10.23820083204779950667791926788, 12.05233203066631695681348740682, 12.97522619176627902992971949068

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