Properties

Label 2-105-3.2-c2-0-0
Degree $2$
Conductor $105$
Sign $-0.0588 + 0.998i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.57i·2-s + (−2.99 − 0.176i)3-s − 8.76·4-s + 2.23i·5-s + (0.630 − 10.6i)6-s − 2.64·7-s − 17.0i·8-s + (8.93 + 1.05i)9-s − 7.98·10-s − 12.0i·11-s + (26.2 + 1.54i)12-s − 12.7·13-s − 9.45i·14-s + (0.394 − 6.69i)15-s + 25.7·16-s + 22.2i·17-s + ⋯
L(s)  = 1  + 1.78i·2-s + (−0.998 − 0.0588i)3-s − 2.19·4-s + 0.447i·5-s + (0.105 − 1.78i)6-s − 0.377·7-s − 2.12i·8-s + (0.993 + 0.117i)9-s − 0.798·10-s − 1.09i·11-s + (2.18 + 0.128i)12-s − 0.978·13-s − 0.675i·14-s + (0.0263 − 0.446i)15-s + 1.61·16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0588 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0588 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.0588 + 0.998i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ -0.0588 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.203244 - 0.215573i\)
\(L(\frac12)\) \(\approx\) \(0.203244 - 0.215573i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.99 + 0.176i)T \)
5 \( 1 - 2.23iT \)
7 \( 1 + 2.64T \)
good2 \( 1 - 3.57iT - 4T^{2} \)
11 \( 1 + 12.0iT - 121T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 - 22.2iT - 289T^{2} \)
19 \( 1 + 28.9T + 361T^{2} \)
23 \( 1 - 21.9iT - 529T^{2} \)
29 \( 1 + 11.4iT - 841T^{2} \)
31 \( 1 + 4.93T + 961T^{2} \)
37 \( 1 + 36.7T + 1.36e3T^{2} \)
41 \( 1 - 57.6iT - 1.68e3T^{2} \)
43 \( 1 - 57.7T + 1.84e3T^{2} \)
47 \( 1 + 15.6iT - 2.20e3T^{2} \)
53 \( 1 - 91.2iT - 2.80e3T^{2} \)
59 \( 1 + 34.3iT - 3.48e3T^{2} \)
61 \( 1 - 71.1T + 3.72e3T^{2} \)
67 \( 1 - 20.1T + 4.48e3T^{2} \)
71 \( 1 - 69.7iT - 5.04e3T^{2} \)
73 \( 1 + 96.8T + 5.32e3T^{2} \)
79 \( 1 + 84.2T + 6.24e3T^{2} \)
83 \( 1 - 20.4iT - 6.88e3T^{2} \)
89 \( 1 - 1.65iT - 7.92e3T^{2} \)
97 \( 1 - 29.7T + 9.40e3T^{2} \)
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

−14.62353378892697940779579075655, −13.44873377577564875967225140803, −12.54970661728271409760106934498, −11.02208437116924936666313121120, −9.885136780283281338605136003132, −8.493688642136917351183169089067, −7.32511413264275446108590153283, −6.31982720173690976455582159090, −5.65634101622987438595009258935, −4.17977299678763279107470124516, 0.24324036504492440906232241136, 2.20852912556979091778262547154, 4.24026137796472148641230532607, 5.09172317689506899969917633046, 7.02948391173237768753654299599, 9.011679172117727666874950262740, 9.954860461321599851468966359499, 10.65347575946798157823684473147, 11.85715417421450242722034520085, 12.45959023172307013762901266621

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