↑

Properties

Label 2-70-7.5-c2-0-1
Degree $2$
Conductor $70$
Sign $0.999 - 0.0305i$
Analytic cond. $1.90736$
Root an. cond. $1.38107$
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  + (−0.707 + 1.22i)2-s + (3.74 − 2.16i)3-s + (−0.999 − 1.73i)4-s + (1.93 + 1.11i)5-s + 6.11i·6-s + (−1.09 − 6.91i)7-s + 2.82·8-s + (4.85 − 8.40i)9-s + (−2.73 + 1.58i)10-s + (3.49 + 6.04i)11-s + (−7.49 − 4.32i)12-s + 22.0i·13-s + (9.24 + 3.54i)14-s + 9.66·15-s + (−2.00 + 3.46i)16-s + (0.711 − 0.410i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (1.24 − 0.720i)3-s + (−0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + 1.01i·6-s + (−0.156 − 0.987i)7-s + 0.353·8-s + (0.538 − 0.933i)9-s + (−0.273 + 0.158i)10-s + (0.317 + 0.549i)11-s + (−0.624 − 0.360i)12-s + 1.69i·13-s + (0.660 + 0.253i)14-s + 0.644·15-s + (−0.125 + 0.216i)16-s + (0.0418 − 0.0241i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0305i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0305i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ 0.999 - 0.0305i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.46833 + 0.0224063i\)
\(L(\frac12)\) \(\approx\) \(1.46833 + 0.0224063i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
5 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 + (1.09 + 6.91i)T \)
good3 \( 1 + (-3.74 + 2.16i)T + (4.5 - 7.79i)T^{2} \)
11 \( 1 + (-3.49 - 6.04i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 22.0iT - 169T^{2} \)
17 \( 1 + (-0.711 + 0.410i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (32.0 + 18.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-2.84 + 4.92i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 23.1T + 841T^{2} \)
31 \( 1 + (6.45 - 3.72i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (19.2 - 33.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 59.8iT - 1.68e3T^{2} \)
43 \( 1 - 30.2T + 1.84e3T^{2} \)
47 \( 1 + (-55.1 - 31.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-9.61 - 16.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-67.4 + 38.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (13.3 + 7.68i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (22.8 + 39.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 98.8T + 5.04e3T^{2} \)
73 \( 1 + (-16.3 + 9.42i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (25.2 - 43.7i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 34.9iT - 6.88e3T^{2} \)
89 \( 1 + (71.6 + 41.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 177. iT - 9.40e3T^{2} \)
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

−14.26715432977259164908976469756, −13.78048838289570545712060067162, −12.73135225955653751979938405256, −10.88369640600896252538133441453, −9.472978310817169939693048497197, −8.666267594708556775048391992670, −7.22384652667118140723347552809, −6.68966402056371359931662076411, −4.20470696881117187258020412166, −1.98981142733855043247127359241, 2.45465577486922696483534667078, 3.73989449701530455698228778398, 5.69627145048351312743247367414, 8.135462293074760064157476172863, 8.792118044469118127315346190358, 9.778093625311101696894755059421, 10.76658109789936441966744610562, 12.41272018060757243181393077913, 13.26586969567346946868352649847, 14.61407520247882016723137149928

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