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Properties

Label 2-70-7.2-c1-0-1
Degree $2$
Conductor $70$
Sign $0.991 + 0.126i$
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  + (0.5 − 0.866i)2-s + (1 + 1.73i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 1.99·6-s + (−2 − 1.73i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−1.5 − 2.59i)11-s + (0.999 − 1.73i)12-s − 13-s + (−2.5 + 0.866i)14-s − 1.99·15-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.577 + 0.999i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.816·6-s + (−0.755 − 0.654i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.452 − 0.783i)11-s + (0.288 − 0.499i)12-s − 0.277·13-s + (−0.668 + 0.231i)14-s − 0.516·15-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11281 - 0.0706192i\)
\(L(\frac12)\) \(\approx\) \(1.11281 - 0.0706192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2 + 1.73i)T \)
good3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−14.61777141842794189768454698025, −13.71089637602596785375825156220, −12.59224917444173072857038973396, −11.12291794042038496292391479291, −10.19095008837039158422829506066, −9.433724438513604545475250100804, −7.84833271908281547941958143290, −5.95989562751509034618273960297, −4.06859797742588904077155461716, −3.20395034731392218686573802636, 2.73046000169629407206692505395, 4.92336720086814895226941027035, 6.55221263625625154346273123197, 7.60422602125420421213833691831, 8.617441261340900804509559804041, 9.948938616610033244479183380716, 12.24788227275347666382833646586, 12.50121732159152632150920036533, 13.68579527590435481843316327750, 14.57781183609616189848107520147

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