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Properties

Label 2-70-35.24-c4-0-6
Degree $2$
Conductor $70$
Sign $0.995 + 0.0996i$
Analytic cond. $7.23589$
Root an. cond. $2.68996$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 + 1.41i)2-s + (−6.27 + 10.8i)3-s + (3.99 − 6.92i)4-s + (10.1 − 22.8i)5-s − 35.4i·6-s + (−32.1 − 37.0i)7-s + 22.6i·8-s + (−38.1 − 66.0i)9-s + (7.48 + 70.3i)10-s + (3.60 − 6.23i)11-s + (50.1 + 86.8i)12-s + 292.·13-s + (130. + 45.2i)14-s + (184. + 253. i)15-s + (−32.0 − 55.4i)16-s + (61.4 − 106. i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.696 + 1.20i)3-s + (0.249 − 0.433i)4-s + (0.405 − 0.914i)5-s − 0.985i·6-s + (−0.655 − 0.755i)7-s + 0.353i·8-s + (−0.470 − 0.815i)9-s + (0.0748 + 0.703i)10-s + (0.0297 − 0.0515i)11-s + (0.348 + 0.603i)12-s + 1.73·13-s + (0.668 + 0.230i)14-s + (0.820 + 1.12i)15-s + (−0.125 − 0.216i)16-s + (0.212 − 0.368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0996i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 + 0.0996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.995 + 0.0996i$
Analytic conductor: \(7.23589\)
Root analytic conductor: \(2.68996\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :2),\ 0.995 + 0.0996i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.911709 - 0.0455295i\)
\(L(\frac12)\) \(\approx\) \(0.911709 - 0.0455295i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.44 - 1.41i)T \)
5 \( 1 + (-10.1 + 22.8i)T \)
7 \( 1 + (32.1 + 37.0i)T \)
good3 \( 1 + (6.27 - 10.8i)T + (-40.5 - 70.1i)T^{2} \)
11 \( 1 + (-3.60 + 6.23i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 - 292.T + 2.85e4T^{2} \)
17 \( 1 + (-61.4 + 106. i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (75.8 - 43.7i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-445. + 257. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 931.T + 7.07e5T^{2} \)
31 \( 1 + (12.4 + 7.20i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-1.49e3 + 864. i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 2.54e3iT - 2.82e6T^{2} \)
43 \( 1 + 410. iT - 3.41e6T^{2} \)
47 \( 1 + (1.51e3 + 2.63e3i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (1.72e3 + 994. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (3.38e3 + 1.95e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-4.13e3 + 2.38e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-6.62e3 - 3.82e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 1.84e3T + 2.54e7T^{2} \)
73 \( 1 + (5.29e3 - 9.17e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-1.89e3 - 3.27e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 3.66e3T + 4.74e7T^{2} \)
89 \( 1 + (-129. + 74.8i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 1.27e4T + 8.85e7T^{2} \)
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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

−14.01277364543042237791410625889, −12.88325269517403592501926447259, −11.25867486835523833138275347022, −10.37950405112358338912508403571, −9.491937806757045386553496984803, −8.451542875538921730676429094434, −6.52147102143053912159685548114, −5.35339623134345912062348451916, −3.97834574798310856842223374136, −0.74643862162899675193814615540, 1.34855670422505051453066158939, 3.01707259861165401624897939658, 6.08878481041212899114479629058, 6.58359277409821621745026751728, 8.054490633490517914272212846008, 9.469320214490497315780895143282, 10.85267794015707123696074389351, 11.62757666687077059463127036316, 12.80980147566442869982317979955, 13.52246437138507783488986733046

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