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Properties

Label 2-70-5.2-c2-0-5
Degree $2$
Conductor $70$
Sign $0.999 + 0.0303i$
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 + i)2-s + (3.89 − 3.89i)3-s + 2i·4-s + (−2.75 + 4.17i)5-s + 7.79·6-s + (−1.87 − 1.87i)7-s + (−2 + 2i)8-s − 21.3i·9-s + (−6.92 + 1.41i)10-s + 6.78·11-s + (7.79 + 7.79i)12-s + (−15.7 + 15.7i)13-s − 3.74i·14-s + (5.51 + 27.0i)15-s − 4·16-s + (−5.59 − 5.59i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (1.29 − 1.29i)3-s + 0.5i·4-s + (−0.551 + 0.834i)5-s + 1.29·6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s − 2.37i·9-s + (−0.692 + 0.141i)10-s + 0.617·11-s + (0.649 + 0.649i)12-s + (−1.21 + 1.21i)13-s − 0.267i·14-s + (0.367 + 1.80i)15-s − 0.250·16-s + (−0.328 − 0.328i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0303i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0303i$
Analytic conductor: \(1.90736\)
Root analytic conductor: \(1.38107\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1),\ 0.999 + 0.0303i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.95451 - 0.0297059i\)
\(L(\frac12)\) \(\approx\) \(1.95451 - 0.0297059i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 + (2.75 - 4.17i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good3 \( 1 + (-3.89 + 3.89i)T - 9iT^{2} \)
11 \( 1 - 6.78T + 121T^{2} \)
13 \( 1 + (15.7 - 15.7i)T - 169iT^{2} \)
17 \( 1 + (5.59 + 5.59i)T + 289iT^{2} \)
19 \( 1 - 11.3iT - 361T^{2} \)
23 \( 1 + (-11.1 + 11.1i)T - 529iT^{2} \)
29 \( 1 - 5.54iT - 841T^{2} \)
31 \( 1 - 5.36T + 961T^{2} \)
37 \( 1 + (-6.62 - 6.62i)T + 1.36e3iT^{2} \)
41 \( 1 - 56.7T + 1.68e3T^{2} \)
43 \( 1 + (-0.984 + 0.984i)T - 1.84e3iT^{2} \)
47 \( 1 + (50.7 + 50.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (-53.9 + 53.9i)T - 2.80e3iT^{2} \)
59 \( 1 - 7.29iT - 3.48e3T^{2} \)
61 \( 1 - 24.2T + 3.72e3T^{2} \)
67 \( 1 + (-51.1 - 51.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 42.7T + 5.04e3T^{2} \)
73 \( 1 + (37.8 - 37.8i)T - 5.32e3iT^{2} \)
79 \( 1 - 44.8iT - 6.24e3T^{2} \)
83 \( 1 + (-38.9 + 38.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 81.9iT - 7.92e3T^{2} \)
97 \( 1 + (60.8 + 60.8i)T + 9.40e3iT^{2} \)
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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

−14.49273959047398733109074702637, −13.65645500968508254058095994996, −12.49972919828197774498787155928, −11.64416389449839921773924502008, −9.532199953320422934448456514093, −8.260148376435622303679105832053, −7.09263243525793023531138962012, −6.70683305320137999369967626726, −3.93113990083557607799613933068, −2.49997727023490992913163469447, 2.83056806224277567147719747212, 4.12833286679855375664893655685, 5.17058083422883427719907717585, 7.81629164312886624863913016242, 9.041959600229843057880082690707, 9.743958094538831414439900561577, 11.01105449707960471114181664207, 12.44973367141172288768773451785, 13.40911137915999832954926109777, 14.69213048062162587215684215228

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