↑

Properties

Label 2-105-35.12-c1-0-6
Degree $2$
Conductor $105$
Sign $0.891 + 0.453i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.49 − 0.401i)2-s + (0.258 − 0.965i)3-s + (0.346 − 0.200i)4-s + (1.61 + 1.54i)5-s − 1.54i·6-s + (−2.44 − 1.01i)7-s + (−1.75 + 1.75i)8-s + (−0.866 − 0.499i)9-s + (3.03 + 1.66i)10-s + (−2.59 − 4.49i)11-s + (−0.103 − 0.386i)12-s + (3.30 + 3.30i)13-s + (−4.06 − 0.545i)14-s + (1.91 − 1.15i)15-s + (−2.32 + 4.01i)16-s + (0.0194 + 0.00519i)17-s + ⋯
L(s)  = 1  + (1.05 − 0.283i)2-s + (0.149 − 0.557i)3-s + (0.173 − 0.100i)4-s + (0.722 + 0.691i)5-s − 0.632i·6-s + (−0.922 − 0.385i)7-s + (−0.619 + 0.619i)8-s + (−0.288 − 0.166i)9-s + (0.960 + 0.527i)10-s + (−0.782 − 1.35i)11-s + (−0.0299 − 0.111i)12-s + (0.917 + 0.917i)13-s + (−1.08 − 0.145i)14-s + (0.493 − 0.299i)15-s + (−0.580 + 1.00i)16-s + (0.00470 + 0.00126i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.891 + 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54458 - 0.370212i\)
\(L(\frac12)\) \(\approx\) \(1.54458 - 0.370212i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-1.61 - 1.54i)T \)
7 \( 1 + (2.44 + 1.01i)T \)
good2 \( 1 + (-1.49 + 0.401i)T + (1.73 - i)T^{2} \)
11 \( 1 + (2.59 + 4.49i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.30 - 3.30i)T + 13iT^{2} \)
17 \( 1 + (-0.0194 - 0.00519i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.24 - 2.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.601 - 2.24i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 + (-5.69 + 3.28i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.66 + 0.714i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 3.68iT - 41T^{2} \)
43 \( 1 + (2.79 - 2.79i)T - 43iT^{2} \)
47 \( 1 + (-0.303 - 1.13i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (4.60 + 1.23i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.222 - 0.385i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.18 - 0.684i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.52 - 5.70i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.14T + 71T^{2} \)
73 \( 1 + (-1.91 + 7.15i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.47 - 2.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.77 - 3.77i)T + 83iT^{2} \)
89 \( 1 + (1.91 - 3.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.5 - 10.5i)T - 97iT^{2} \)
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

−13.49683037378287132195231062085, −13.21084238163971463552436416764, −11.76283270002127555263581751253, −10.83134358305765377206867320351, −9.464265917850366377876492372362, −8.136292919111214686180317886299, −6.40976634919966151509117212518, −5.84500829723210984134611328745, −3.80075476120480464918338634239, −2.66458456791769646682734844146, 3.00691975276987936494930720071, 4.64198109685902315384889271854, 5.48857277638717549341468147687, 6.64134178507212222106184225947, 8.610185670593812637147061367719, 9.621087099807974612963223885184, 10.44428663920326931609559917307, 12.44688931277085937940891115346, 12.87857102642514110036914990482, 13.70134691384441346164626961987

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