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Properties

Label 2-70-35.4-c1-0-0
Degree $2$
Conductor $70$
Sign $0.830 - 0.556i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (1.86 + 1.23i)5-s + (1.73 + 2i)7-s + 0.999i·8-s + (−1.5 − 2.59i)9-s + (−2.23 − 0.133i)10-s + (−1.5 + 2.59i)11-s − 5i·13-s + (−2.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (2.59 + 1.5i)18-s + (−2.5 − 4.33i)19-s + (1.99 − i)20-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.834 + 0.550i)5-s + (0.654 + 0.755i)7-s + 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.705 − 0.0423i)10-s + (−0.452 + 0.783i)11-s − 1.38i·13-s + (−0.668 − 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + (0.612 + 0.353i)18-s + (−0.573 − 0.993i)19-s + (0.447 − 0.223i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.830 - 0.556i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.830 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.745488 + 0.226690i\)
\(L(\frac12)\) \(\approx\) \(0.745488 + 0.226690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-1.86 - 1.23i)T \)
7 \( 1 + (-1.73 - 2i)T \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.06 - 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + (-6.06 + 3.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.79 + 4.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-13.8 - 8i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.11820618583151164893421984566, −13.99633763785603417426204265515, −12.59866913782021259785722470759, −11.29114547768061610498030367527, −10.15172993684019108314376753627, −9.130416190934001159258178329616, −7.903820222778613211112110032568, −6.42033470391414466067033464248, −5.31991896541835435014788028969, −2.50548909992439682869609315126, 1.98793399665740071309754400575, 4.48972306656910139104697797137, 6.18661944655002757176010237946, 7.963653473364360370948674694096, 8.806108920344191086922236346895, 10.25321854300404979655528034078, 11.00288960029022323344523879439, 12.30768741589070514300946397983, 13.74891114599147553475502777186, 14.14616756403520618343136341797

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