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Properties

Label 2-70-7.2-c3-0-6
Degree $2$
Conductor $70$
Sign $-0.999 + 0.00346i$
Analytic cond. $4.13013$
Root an. cond. $2.03227$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−2.89 − 5.00i)3-s + (−1.99 − 3.46i)4-s + (−2.5 + 4.33i)5-s − 11.5·6-s + (−15.4 − 10.2i)7-s − 7.99·8-s + (−3.21 + 5.57i)9-s + (5 + 8.66i)10-s + (10.2 + 17.6i)11-s + (−11.5 + 20.0i)12-s − 53.1·13-s + (−33.1 + 16.5i)14-s + 28.9·15-s + (−8 + 13.8i)16-s + (−13.8 − 24.0i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.556 − 0.963i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 0.786·6-s + (−0.834 − 0.550i)7-s − 0.353·8-s + (−0.119 + 0.206i)9-s + (0.158 + 0.273i)10-s + (0.280 + 0.485i)11-s + (−0.278 + 0.481i)12-s − 1.13·13-s + (−0.632 + 0.316i)14-s + 0.497·15-s + (−0.125 + 0.216i)16-s + (−0.197 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00346i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $-0.999 + 0.00346i$
Analytic conductor: \(4.13013\)
Root analytic conductor: \(2.03227\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :3/2),\ -0.999 + 0.00346i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00160137 - 0.923536i\)
\(L(\frac12)\) \(\approx\) \(0.00160137 - 0.923536i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (15.4 + 10.2i)T \)
good3 \( 1 + (2.89 + 5.00i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-10.2 - 17.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 53.1T + 2.19e3T^{2} \)
17 \( 1 + (13.8 + 24.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-71.5 + 123. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-100. + 173. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 113.T + 2.43e4T^{2} \)
31 \( 1 + (-52.7 - 91.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-0.693 + 1.20i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 226.T + 6.89e4T^{2} \)
43 \( 1 - 268.T + 7.95e4T^{2} \)
47 \( 1 + (13.8 - 24.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (37.1 + 64.3i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (332. + 576. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (254. - 441. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-490. - 849. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 144.T + 3.57e5T^{2} \)
73 \( 1 + (367. + 637. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (73.0 - 126. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 712.T + 5.71e5T^{2} \)
89 \( 1 + (-303. + 525. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 771.T + 9.12e5T^{2} \)
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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

−13.33585852829796997319643677360, −12.50214739548963833430783949936, −11.71879321570564223189305359366, −10.49122771045833041961789014277, −9.332167921517704347151053935509, −7.20204642250059202803497832186, −6.61026142962725212377369955597, −4.72480482423197285902779222379, −2.79988302990728765627727589898, −0.58064216224983414772452059515, 3.54360304949152139130291387222, 5.00518063368955497705965504742, 5.97644566124971808684044561295, 7.62807872795419079366606841240, 9.185049343648409287992358982274, 10.05609632055212869343064055345, 11.60993832151310474638304203142, 12.51119346613984989543994438091, 13.76835347568955514204982002610, 15.11864936405982866265825247452

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