↑

Properties

Label 2-105-35.13-c3-0-21
Degree $2$
Conductor $105$
Sign $-0.501 + 0.864i$
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  + (3.82 − 3.82i)2-s + (2.12 − 2.12i)3-s − 21.3i·4-s + (10.1 + 4.60i)5-s − 16.2i·6-s + (−6.06 + 17.4i)7-s + (−51.0 − 51.0i)8-s − 8.99i·9-s + (56.6 − 21.3i)10-s + 38.1·11-s + (−45.2 − 45.2i)12-s + (−62.7 + 62.7i)13-s + (43.7 + 90.2i)14-s + (31.3 − 11.8i)15-s − 220.·16-s + (15.5 + 15.5i)17-s + ⋯
L(s)  = 1  + (1.35 − 1.35i)2-s + (0.408 − 0.408i)3-s − 2.66i·4-s + (0.911 + 0.411i)5-s − 1.10i·6-s + (−0.327 + 0.944i)7-s + (−2.25 − 2.25i)8-s − 0.333i·9-s + (1.79 − 0.676i)10-s + 1.04·11-s + (−1.08 − 1.08i)12-s + (−1.33 + 1.33i)13-s + (0.835 + 1.72i)14-s + (0.540 − 0.204i)15-s − 3.43·16-s + (0.221 + 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.501 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.501 + 0.864i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.501 + 0.864i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.82467 - 3.16868i\)
\(L(\frac12)\) \(\approx\) \(1.82467 - 3.16868i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.1 - 4.60i)T \)
7 \( 1 + (6.06 - 17.4i)T \)
good2 \( 1 + (-3.82 + 3.82i)T - 8iT^{2} \)
11 \( 1 - 38.1T + 1.33e3T^{2} \)
13 \( 1 + (62.7 - 62.7i)T - 2.19e3iT^{2} \)
17 \( 1 + (-15.5 - 15.5i)T + 4.91e3iT^{2} \)
19 \( 1 + 25.6T + 6.85e3T^{2} \)
23 \( 1 + (21.8 + 21.8i)T + 1.21e4iT^{2} \)
29 \( 1 + 82.7iT - 2.43e4T^{2} \)
31 \( 1 + 197. iT - 2.97e4T^{2} \)
37 \( 1 + (-111. + 111. i)T - 5.06e4iT^{2} \)
41 \( 1 - 28.5iT - 6.89e4T^{2} \)
43 \( 1 + (-76.5 - 76.5i)T + 7.95e4iT^{2} \)
47 \( 1 + (-265. - 265. i)T + 1.03e5iT^{2} \)
53 \( 1 + (124. + 124. i)T + 1.48e5iT^{2} \)
59 \( 1 + 480.T + 2.05e5T^{2} \)
61 \( 1 + 246. iT - 2.26e5T^{2} \)
67 \( 1 + (179. - 179. i)T - 3.00e5iT^{2} \)
71 \( 1 - 142.T + 3.57e5T^{2} \)
73 \( 1 + (-206. + 206. i)T - 3.89e5iT^{2} \)
79 \( 1 - 982. iT - 4.93e5T^{2} \)
83 \( 1 + (433. - 433. i)T - 5.71e5iT^{2} \)
89 \( 1 + 864.T + 7.04e5T^{2} \)
97 \( 1 + (814. + 814. i)T + 9.12e5iT^{2} \)
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

−12.79299411959288599770065996461, −12.12093341796982440980647256597, −11.23802250476463973838544321443, −9.698356384773358736448392800863, −9.331417785894943396668401447079, −6.66639744122786663972267442166, −5.81373297464449836033684607453, −4.28103612900494417221967145224, −2.67914958410620036471398445536, −1.85559384586650155311307335972, 3.09808055085040259235368779997, 4.44761434637586317442612552734, 5.45794887832912502818025233851, 6.71033119398583343134276488156, 7.72971387828407664856398660951, 9.040665188268471455031110863926, 10.28188413794803867509691043990, 12.21563596826032152639311406363, 12.98542243868839892839424472122, 13.93292172192942668090204660549

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