↑

Properties

Label 2-70-5.2-c2-0-0
Degree $2$
Conductor $70$
Sign $-0.678 - 0.734i$
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  + (1 + i)2-s + (−2.48 + 2.48i)3-s + 2i·4-s + (−4.90 + 0.953i)5-s − 4.96·6-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s − 3.32i·9-s + (−5.86 − 3.95i)10-s + 14.2·11-s + (−4.96 − 4.96i)12-s + (−6.32 + 6.32i)13-s + 3.74i·14-s + (9.81 − 14.5i)15-s − 4·16-s + (23.2 + 23.2i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.827 + 0.827i)3-s + 0.5i·4-s + (−0.981 + 0.190i)5-s − 0.827·6-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.369i·9-s + (−0.586 − 0.395i)10-s + 1.29·11-s + (−0.413 − 0.413i)12-s + (−0.486 + 0.486i)13-s + 0.267i·14-s + (0.654 − 0.970i)15-s − 0.250·16-s + (1.36 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ -0.678 - 0.734i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.421234 + 0.962156i\)
\(L(\frac12)\) \(\approx\) \(0.421234 + 0.962156i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (4.90 - 0.953i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good3 \( 1 + (2.48 - 2.48i)T - 9iT^{2} \)
11 \( 1 - 14.2T + 121T^{2} \)
13 \( 1 + (6.32 - 6.32i)T - 169iT^{2} \)
17 \( 1 + (-23.2 - 23.2i)T + 289iT^{2} \)
19 \( 1 + 26.8iT - 361T^{2} \)
23 \( 1 + (6.00 - 6.00i)T - 529iT^{2} \)
29 \( 1 - 4.17iT - 841T^{2} \)
31 \( 1 - 7.54T + 961T^{2} \)
37 \( 1 + (9.17 + 9.17i)T + 1.36e3iT^{2} \)
41 \( 1 + 26.3T + 1.68e3T^{2} \)
43 \( 1 + (-27.7 + 27.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-38.8 - 38.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (50.2 - 50.2i)T - 2.80e3iT^{2} \)
59 \( 1 - 17.3iT - 3.48e3T^{2} \)
61 \( 1 - 3.10T + 3.72e3T^{2} \)
67 \( 1 + (36.8 + 36.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 127.T + 5.04e3T^{2} \)
73 \( 1 + (-65.5 + 65.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 55.8iT - 6.24e3T^{2} \)
83 \( 1 + (-89.9 + 89.9i)T - 6.88e3iT^{2} \)
89 \( 1 + 92.3iT - 7.92e3T^{2} \)
97 \( 1 + (7.00 + 7.00i)T + 9.40e3iT^{2} \)
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

−15.04337236663730360028748260025, −14.14168820134105313937338315738, −12.33800599195679629594943157964, −11.67837938984228851479055081748, −10.66843789062643195186724386161, −9.107725466386170352032753170411, −7.65264760700229210055387055631, −6.26697193507679332053001248307, −4.85000530926039546642826186812, −3.79961641359911836636452793711, 0.981115213727887376065048548522, 3.70407383795107388508117954515, 5.30469084806088616026557205858, 6.75940537731982721545060996356, 7.908006652376378201054535864105, 9.746615225134552501541728602074, 11.24147148577368522484961993408, 12.11563776759126347658679624225, 12.36604354096904021049928648561, 13.99578772543077723983752038347

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