↑

Properties

Label 2-70-35.4-c3-0-6
Degree $2$
Conductor $70$
Sign $0.995 + 0.0894i$
Analytic cond. $4.13013$
Root an. cond. $2.03227$
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  + (1.73 − i)2-s + (2.65 + 1.53i)3-s + (1.99 − 3.46i)4-s + (10.0 + 4.89i)5-s + 6.13·6-s + (−1.78 + 18.4i)7-s − 7.99i·8-s + (−8.79 − 15.2i)9-s + (22.3 − 1.56i)10-s + (24.7 − 42.8i)11-s + (10.6 − 6.13i)12-s + 44.6i·13-s + (15.3 + 33.7i)14-s + (19.1 + 28.4i)15-s + (−8 − 13.8i)16-s + (−34.0 − 19.6i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.511 + 0.295i)3-s + (0.249 − 0.433i)4-s + (0.898 + 0.438i)5-s + 0.417·6-s + (−0.0961 + 0.995i)7-s − 0.353i·8-s + (−0.325 − 0.564i)9-s + (0.705 − 0.0495i)10-s + (0.678 − 1.17i)11-s + (0.255 − 0.147i)12-s + 0.951i·13-s + (0.293 + 0.643i)14-s + (0.330 + 0.489i)15-s + (−0.125 − 0.216i)16-s + (−0.485 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.995 + 0.0894i$
Analytic conductor: \(4.13013\)
Root analytic conductor: \(2.03227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :3/2),\ 0.995 + 0.0894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.49917 - 0.111970i\)
\(L(\frac12)\) \(\approx\) \(2.49917 - 0.111970i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.73 + i)T \)
5 \( 1 + (-10.0 - 4.89i)T \)
7 \( 1 + (1.78 - 18.4i)T \)
good3 \( 1 + (-2.65 - 1.53i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-24.7 + 42.8i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 44.6iT - 2.19e3T^{2} \)
17 \( 1 + (34.0 + 19.6i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (28.1 + 48.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (79.3 - 45.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 281.T + 2.43e4T^{2} \)
31 \( 1 + (35.2 - 61.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (170. - 98.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 399.T + 6.89e4T^{2} \)
43 \( 1 + 203. iT - 7.95e4T^{2} \)
47 \( 1 + (59.3 - 34.2i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-534. - 308. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-240. + 416. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-11.7 - 20.3i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-218. - 126. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 835.T + 3.57e5T^{2} \)
73 \( 1 + (-223. - 128. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (386. + 670. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.34e3iT - 5.71e5T^{2} \)
89 \( 1 + (-650. - 1.12e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 323. iT - 9.12e5T^{2} \)
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

−14.19818675925090135878182641617, −13.35012307996190475163302777564, −11.91851554637541335469047629630, −11.05284986085211484978242709899, −9.420345861686204133037050227923, −8.918382968181064001021407479486, −6.56712639643848836436051278148, −5.63702968791540713053241735471, −3.64614575035342084995661134578, −2.26914873729833351414862059924, 2.03744396426156985242424172890, 4.10303268553142669073137346685, 5.61036564282988678331970767945, 7.05107718837679351880680152468, 8.175283746750857215612762589720, 9.633001096404053816117750445390, 10.82607730200813774329699442167, 12.60043528692015263238159063047, 13.20046297047075137180200955604, 14.17299261399555987823266404464

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