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Properties

Label 2-70-35.12-c1-0-1
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: $\chi_{70} (47, \cdot )$
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

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

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