Properties

Label 2-2070-23.22-c2-0-46
Degree $2$
Conductor $2070$
Sign $0.491 + 0.870i$
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 − 7.10i·7-s − 2.82·8-s + 3.16i·10-s + 11.2i·11-s + 20.0·13-s + 10.0i·14-s + 4.00·16-s + 1.63i·17-s − 29.4i·19-s − 4.47i·20-s − 15.9i·22-s + (−20.0 + 11.3i)23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s − 0.447i·5-s − 1.01i·7-s − 0.353·8-s + 0.316i·10-s + 1.02i·11-s + 1.54·13-s + 0.717i·14-s + 0.250·16-s + 0.0959i·17-s − 1.54i·19-s − 0.223i·20-s − 0.724i·22-s + (−0.870 + 0.491i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.491 + 0.870i$
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: \((2,\ 2070,\ (\ :1),\ 0.491 + 0.870i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.543251655\)
\(L(\frac12)\) \(\approx\) \(1.543251655\)
\(L(2)\) 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 \( 1 + (20.0 - 11.3i)T \)
good7 \( 1 + 7.10iT - 49T^{2} \)
11 \( 1 - 11.2iT - 121T^{2} \)
13 \( 1 - 20.0T + 169T^{2} \)
17 \( 1 - 1.63iT - 289T^{2} \)
19 \( 1 + 29.4iT - 361T^{2} \)
29 \( 1 - 50.3T + 841T^{2} \)
31 \( 1 - 11.1T + 961T^{2} \)
37 \( 1 + 40.5iT - 1.36e3T^{2} \)
41 \( 1 - 7.24T + 1.68e3T^{2} \)
43 \( 1 - 71.7iT - 1.84e3T^{2} \)
47 \( 1 - 6.40T + 2.20e3T^{2} \)
53 \( 1 + 20.4iT - 2.80e3T^{2} \)
59 \( 1 - 65.8T + 3.48e3T^{2} \)
61 \( 1 - 37.7iT - 3.72e3T^{2} \)
67 \( 1 - 124. iT - 4.48e3T^{2} \)
71 \( 1 + 43.5T + 5.04e3T^{2} \)
73 \( 1 - 48.1T + 5.32e3T^{2} \)
79 \( 1 + 101. iT - 6.24e3T^{2} \)
83 \( 1 - 102. iT - 6.88e3T^{2} \)
89 \( 1 + 9.63iT - 7.92e3T^{2} \)
97 \( 1 + 143. iT - 9.40e3T^{2} \)
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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

−8.736727919714423949608290264756, −8.168191282800868504833098065415, −7.26479811749690336736489129537, −6.70162049680144554076689706394, −5.78953911274273433479986729202, −4.58618060583428413051437961131, −3.98370503742171674298022158554, −2.72752436647597891457898565228, −1.46727300270508758068319609885, −0.65111245857275797809021864325, 0.930548317173473956013968382010, 2.09604418328398600738167707045, 3.11324914779772748074257417683, 3.88915075987919366201954465695, 5.37978359641737342781227992654, 6.21589045476311167069866664639, 6.43591484427188270586596744934, 7.84774010586838870491697504864, 8.462539875901069473666986494611, 8.764534681720985903665019522447

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