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Properties

Label 2-70-35.3-c1-0-3
Degree $2$
Conductor $70$
Sign $0.814 + 0.579i$
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  + (0.965 + 0.258i)2-s + (−0.752 − 2.80i)3-s + (0.866 + 0.499i)4-s + (−1.38 + 1.75i)5-s − 2.90i·6-s + (2.58 − 0.559i)7-s + (0.707 + 0.707i)8-s + (−4.71 + 2.72i)9-s + (−1.79 + 1.33i)10-s + (−1.83 + 3.17i)11-s + (0.752 − 2.80i)12-s + (−0.830 + 0.830i)13-s + (2.64 + 0.128i)14-s + (5.97 + 2.55i)15-s + (0.500 + 0.866i)16-s + (−0.761 + 0.204i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.434 − 1.62i)3-s + (0.433 + 0.249i)4-s + (−0.618 + 0.785i)5-s − 1.18i·6-s + (0.977 − 0.211i)7-s + (0.249 + 0.249i)8-s + (−1.57 + 0.908i)9-s + (−0.566 + 0.423i)10-s + (−0.553 + 0.958i)11-s + (0.217 − 0.810i)12-s + (−0.230 + 0.230i)13-s + (0.706 + 0.0343i)14-s + (1.54 + 0.660i)15-s + (0.125 + 0.216i)16-s + (−0.184 + 0.0494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70\)    =    \(2 \cdot 5 \cdot 7\)
Sign: $0.814 + 0.579i$
Analytic conductor: \(0.558952\)
Root analytic conductor: \(0.747631\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{70} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 70,\ (\ :1/2),\ 0.814 + 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04894 - 0.335098i\)
\(L(\frac12)\) \(\approx\) \(1.04894 - 0.335098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (1.38 - 1.75i)T \)
7 \( 1 + (-2.58 + 0.559i)T \)
good3 \( 1 + (0.752 + 2.80i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.83 - 3.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.830 - 0.830i)T - 13iT^{2} \)
17 \( 1 + (0.761 - 0.204i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.09 + 1.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.21 + 4.54i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + (-0.0359 - 0.0207i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.248 + 0.0664i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.98iT - 41T^{2} \)
43 \( 1 + (0.474 + 0.474i)T + 43iT^{2} \)
47 \( 1 + (-1.65 + 6.18i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.64 - 2.04i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.35 - 9.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.72 + 0.996i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.71 + 6.39i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + (-2.55 - 9.52i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-11.6 + 6.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.73 + 9.73i)T - 83iT^{2} \)
89 \( 1 + (0.715 + 1.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.16 - 3.16i)T + 97iT^{2} \)
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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

−14.47667476853707401149726739163, −13.45817745930555453782424707153, −12.42715004887133284620127890243, −11.63871783590518160053304016010, −10.71184812814368679736835084195, −8.081657861748679788956089455972, −7.32803982070522149612718961502, −6.44048752257092059212877618433, −4.72662409090463093984542103532, −2.30907554031432326785527711984, 3.62379563879061456016643983443, 4.85172990646947505548413845891, 5.55443848664942662183933377186, 8.052078582739475480742819099364, 9.263553030552495753115524753862, 10.74742261190635295711086261682, 11.31735970777846074363123825103, 12.41294315573244636232482909694, 13.93310362277067837970281487176, 15.09223353443349947171091987687

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