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Properties

Label 2-70-35.24-c2-0-4
Degree $2$
Conductor $70$
Sign $0.794 + 0.607i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.170 − 0.294i)3-s + (0.999 − 1.73i)4-s + (4.97 − 0.495i)5-s − 0.481i·6-s + (−6.56 − 2.43i)7-s − 2.82i·8-s + (4.44 + 7.69i)9-s + (5.74 − 4.12i)10-s + (3.03 − 5.25i)11-s + (−0.340 − 0.589i)12-s − 3.53·13-s + (−9.75 + 1.65i)14-s + (0.700 − 1.55i)15-s + (−2.00 − 3.46i)16-s + (−13.2 + 23.0i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.0566 − 0.0982i)3-s + (0.249 − 0.433i)4-s + (0.995 − 0.0991i)5-s − 0.0801i·6-s + (−0.937 − 0.347i)7-s − 0.353i·8-s + (0.493 + 0.854i)9-s + (0.574 − 0.412i)10-s + (0.275 − 0.477i)11-s + (−0.0283 − 0.0491i)12-s − 0.272·13-s + (−0.697 + 0.118i)14-s + (0.0466 − 0.103i)15-s + (−0.125 − 0.216i)16-s + (−0.781 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ 0.794 + 0.607i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.68760 - 0.571124i\)
\(L(\frac12)\) \(\approx\) \(1.68760 - 0.571124i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
5 \( 1 + (-4.97 + 0.495i)T \)
7 \( 1 + (6.56 + 2.43i)T \)
good3 \( 1 + (-0.170 + 0.294i)T + (-4.5 - 7.79i)T^{2} \)
11 \( 1 + (-3.03 + 5.25i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 3.53T + 169T^{2} \)
17 \( 1 + (13.2 - 23.0i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (27.3 - 15.8i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-12.9 + 7.46i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 32.6T + 841T^{2} \)
31 \( 1 + (-20.0 - 11.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-30.3 + 17.5i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 42.8iT - 1.68e3T^{2} \)
43 \( 1 + 70.0iT - 1.84e3T^{2} \)
47 \( 1 + (15.2 + 26.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-42.7 - 24.6i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (25.7 + 14.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-31.3 + 18.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-71.3 - 41.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 84.8T + 5.04e3T^{2} \)
73 \( 1 + (5.59 - 9.69i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-3.46 - 6.00i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 67.9T + 6.88e3T^{2} \)
89 \( 1 + (-1.68 + 0.974i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 60.3T + 9.40e3T^{2} \)
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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

−14.06045875114873909337067174846, −13.11224322800761265099022484284, −12.66157373879898376343361953454, −10.76493817398804414230165869204, −10.17220044810533370986043320336, −8.736836703720118158898708944661, −6.82078542990664197090683150148, −5.77124034003722377572385536384, −4.06711629843170791511065195881, −2.10499213712235620633202402739, 2.72213287743171029693337976830, 4.60329587432119010411612019520, 6.23833547641928864136574404827, 6.95220149647834572547899660372, 9.115160939444360295420286885546, 9.772649977698953421641138612534, 11.43446527171152950757548782708, 12.84939511037037325626030158994, 13.26536111430288886946533989343, 14.71623386453939084356143223811

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