Properties

Label 2-105-35.27-c3-0-23
Degree $2$
Conductor $105$
Sign $-0.710 - 0.703i$
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

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.811 − 0.811i)2-s + (−2.12 − 2.12i)3-s − 6.68i·4-s + (−4.54 − 10.2i)5-s + 3.44i·6-s + (−15.2 + 10.5i)7-s + (−11.9 + 11.9i)8-s + 8.99i·9-s + (−4.59 + 11.9i)10-s + 47.0·11-s + (−14.1 + 14.1i)12-s + (6.35 + 6.35i)13-s + (20.9 + 3.84i)14-s + (−12.0 + 31.3i)15-s − 34.1·16-s + (−48.7 + 48.7i)17-s + ⋯
L(s)  = 1  + (−0.286 − 0.286i)2-s + (−0.408 − 0.408i)3-s − 0.835i·4-s + (−0.406 − 0.913i)5-s + 0.234i·6-s + (−0.823 + 0.567i)7-s + (−0.526 + 0.526i)8-s + 0.333i·9-s + (−0.145 + 0.378i)10-s + 1.28·11-s + (−0.341 + 0.341i)12-s + (0.135 + 0.135i)13-s + (0.399 + 0.0734i)14-s + (−0.206 + 0.538i)15-s − 0.533·16-s + (−0.696 + 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.710 - 0.703i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.710 - 0.703i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.115707 + 0.281320i\)
\(L(\frac12)\) \(\approx\) \(0.115707 + 0.281320i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.12 + 2.12i)T \)
5 \( 1 + (4.54 + 10.2i)T \)
7 \( 1 + (15.2 - 10.5i)T \)
good2 \( 1 + (0.811 + 0.811i)T + 8iT^{2} \)
11 \( 1 - 47.0T + 1.33e3T^{2} \)
13 \( 1 + (-6.35 - 6.35i)T + 2.19e3iT^{2} \)
17 \( 1 + (48.7 - 48.7i)T - 4.91e3iT^{2} \)
19 \( 1 + 122.T + 6.85e3T^{2} \)
23 \( 1 + (-32.8 + 32.8i)T - 1.21e4iT^{2} \)
29 \( 1 + 111. iT - 2.43e4T^{2} \)
31 \( 1 + 69.1iT - 2.97e4T^{2} \)
37 \( 1 + (227. + 227. i)T + 5.06e4iT^{2} \)
41 \( 1 + 258. iT - 6.89e4T^{2} \)
43 \( 1 + (331. - 331. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-332. + 332. i)T - 1.03e5iT^{2} \)
53 \( 1 + (397. - 397. i)T - 1.48e5iT^{2} \)
59 \( 1 - 124.T + 2.05e5T^{2} \)
61 \( 1 - 390. iT - 2.26e5T^{2} \)
67 \( 1 + (172. + 172. i)T + 3.00e5iT^{2} \)
71 \( 1 - 146.T + 3.57e5T^{2} \)
73 \( 1 + (360. + 360. i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.19e3iT - 4.93e5T^{2} \)
83 \( 1 + (33.8 + 33.8i)T + 5.71e5iT^{2} \)
89 \( 1 + 1.28e3T + 7.04e5T^{2} \)
97 \( 1 + (-212. + 212. i)T - 9.12e5iT^{2} \)
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

−12.45554845829046491710151486255, −11.66720380440000821888970728793, −10.55716441171367019408943119876, −9.213895969806960157776183883351, −8.631032768828402945816100675599, −6.65234094778494325141498395598, −5.82417964488889533242692878240, −4.26144083158521297826299220618, −1.82909213276862651413749330419, −0.19082839769621224518869948171, 3.22657067947557518856226560433, 4.19767090877755762611541880695, 6.59979051792300734350062051321, 6.89766905157486147636918113185, 8.514284652955676293677336689468, 9.624284465321869375338476853069, 10.81593482036174615283231666113, 11.73482194876568427833037336817, 12.76889518212787172815437272228, 13.98077746753951589091105110708

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