Properties

Label 2-70-35.17-c1-0-3
Degree $2$
Conductor $70$
Sign $0.685 + 0.727i$
Analytic cond. $0.558952$
Root an. cond. $0.747631$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 − 0.965i)2-s + (1.95 − 0.523i)3-s + (−0.866 − 0.499i)4-s + (−1.82 + 1.29i)5-s − 2.02i·6-s + (−1.90 + 1.83i)7-s + (−0.707 + 0.707i)8-s + (0.941 − 0.543i)9-s + (0.774 + 2.09i)10-s + (2.01 − 3.49i)11-s + (−1.95 − 0.523i)12-s + (−0.204 − 0.204i)13-s + (1.28 + 2.31i)14-s + (−2.89 + 3.47i)15-s + (0.500 + 0.866i)16-s + (−0.527 − 1.97i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (1.12 − 0.302i)3-s + (−0.433 − 0.249i)4-s + (−0.816 + 0.577i)5-s − 0.825i·6-s + (−0.718 + 0.695i)7-s + (−0.249 + 0.249i)8-s + (0.313 − 0.181i)9-s + (0.244 + 0.663i)10-s + (0.609 − 1.05i)11-s + (−0.563 − 0.151i)12-s + (−0.0568 − 0.0568i)13-s + (0.343 + 0.618i)14-s + (−0.746 + 0.897i)15-s + (0.125 + 0.216i)16-s + (−0.128 − 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.685 + 0.727i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.685 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01775 - 0.439493i\)
\(L(\frac12)\) \(\approx\) \(1.01775 - 0.439493i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (1.82 - 1.29i)T \)
7 \( 1 + (1.90 - 1.83i)T \)
good3 \( 1 + (-1.95 + 0.523i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.01 + 3.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.204 + 0.204i)T + 13iT^{2} \)
17 \( 1 + (0.527 + 1.97i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.10 - 5.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.38 + 1.17i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 7.15iT - 29T^{2} \)
31 \( 1 + (-6.33 - 3.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.19 + 4.46i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.58iT - 41T^{2} \)
43 \( 1 + (4.97 - 4.97i)T - 43iT^{2} \)
47 \( 1 + (0.304 + 0.0815i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.14 - 8.00i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.427 - 0.740i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.99 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.05 + 0.817i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 + (11.1 - 2.98i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.39 + 2.53i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.85 - 3.85i)T + 83iT^{2} \)
89 \( 1 + (-1.53 - 2.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.63 + 6.63i)T - 97iT^{2} \)
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

−14.28096776120621046111758643679, −13.68948295113775847510920849330, −12.27325448783115957607170603931, −11.50745259284858768659497626760, −10.00212779268881526621358399677, −8.805808909188868843766211655383, −7.85323044676230143572932533137, −6.11789369285584648934890376812, −3.72151607898668227848276029120, −2.74262518317713777047060118223, 3.49343293478766282126562683169, 4.58764485028610811102852598859, 6.79736789480212432017013732695, 7.87393551760799619698812387548, 8.992457581677683318599991035069, 9.876278890333592223039078553262, 11.81441685812731266893856370687, 13.00986924943861798589532042553, 13.91890545644486055968246995273, 15.02114838066296149646306805153

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