Properties

Label 2-2070-23.22-c2-0-61
Degree $2$
Conductor $2070$
Sign $-0.292 + 0.956i$
Analytic cond. $56.4034$
Root an. cond. $7.51022$
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  − 1.41·2-s + 2.00·4-s + 2.23i·5-s − 11.8i·7-s − 2.82·8-s − 3.16i·10-s − 20.7i·11-s + 7.75·13-s + 16.7i·14-s + 4.00·16-s + 22.9i·17-s − 21.0i·19-s + 4.47i·20-s + 29.3i·22-s + (21.9 + 6.71i)23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s + 0.447i·5-s − 1.68i·7-s − 0.353·8-s − 0.316i·10-s − 1.88i·11-s + 0.596·13-s + 1.19i·14-s + 0.250·16-s + 1.34i·17-s − 1.10i·19-s + 0.223i·20-s + 1.33i·22-s + (0.956 + 0.292i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.292 + 0.956i$
Analytic conductor: \(56.4034\)
Root analytic conductor: \(7.51022\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2070} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2070,\ (\ :1),\ -0.292 + 0.956i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.440890977\)
\(L(\frac12)\) \(\approx\) \(1.440890977\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
5 \( 1 - 2.23iT \)
23 \( 1 + (-21.9 - 6.71i)T \)
good7 \( 1 + 11.8iT - 49T^{2} \)
11 \( 1 + 20.7iT - 121T^{2} \)
13 \( 1 - 7.75T + 169T^{2} \)
17 \( 1 - 22.9iT - 289T^{2} \)
19 \( 1 + 21.0iT - 361T^{2} \)
29 \( 1 - 38.2T + 841T^{2} \)
31 \( 1 - 33.3T + 961T^{2} \)
37 \( 1 + 49.4iT - 1.36e3T^{2} \)
41 \( 1 - 0.679T + 1.68e3T^{2} \)
43 \( 1 - 7.99iT - 1.84e3T^{2} \)
47 \( 1 - 85.9T + 2.20e3T^{2} \)
53 \( 1 - 66.6iT - 2.80e3T^{2} \)
59 \( 1 + 110.T + 3.48e3T^{2} \)
61 \( 1 + 16.6iT - 3.72e3T^{2} \)
67 \( 1 + 117. iT - 4.48e3T^{2} \)
71 \( 1 + 28.9T + 5.04e3T^{2} \)
73 \( 1 - 31.0T + 5.32e3T^{2} \)
79 \( 1 - 105. iT - 6.24e3T^{2} \)
83 \( 1 + 43.1iT - 6.88e3T^{2} \)
89 \( 1 - 49.1iT - 7.92e3T^{2} \)
97 \( 1 + 69.6iT - 9.40e3T^{2} \)
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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

−8.634094123530411314564893797857, −8.005573167848041308456455042633, −7.22114802544061482916361694364, −6.46071682313037912759112574167, −5.88257012552311644534874626808, −4.46084441816531580566177977000, −3.57555184280955158485295664558, −2.85663145586902031271908072287, −1.19203543565429720840188038744, −0.55970021844600723507728475626, 1.18664174978133481462560904008, 2.23079882969989444874537783202, 2.95393485840776362747179597334, 4.53623599656229171896867985365, 5.15493253214656036664491061722, 6.11902152692400116365692227141, 6.88894866879087311598362436932, 7.78078871282195162531569192493, 8.580361223044125377344657659887, 9.087255664281258043146470224237

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