Properties

Label 2-4140-23.22-c2-0-1
Degree $2$
Conductor $4140$
Sign $-0.956 - 0.292i$
Analytic cond. $112.806$
Root an. cond. $10.6210$
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  + 2.23i·5-s − 8.25i·7-s − 1.54i·11-s − 9.82·13-s + 30.3i·17-s − 35.7i·19-s + (−6.71 + 21.9i)23-s − 5.00·25-s + 26.2·29-s + 20.1·31-s + 18.4·35-s − 29.5i·37-s + 4.35·41-s + 14.5i·43-s − 21.5·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 1.17i·7-s − 0.140i·11-s − 0.755·13-s + 1.78i·17-s − 1.87i·19-s + (−0.292 + 0.956i)23-s − 0.200·25-s + 0.903·29-s + 0.651·31-s + 0.527·35-s − 0.799i·37-s + 0.106·41-s + 0.337i·43-s − 0.458·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.956 - 0.292i$
Analytic conductor: \(112.806\)
Root analytic conductor: \(10.6210\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1),\ -0.956 - 0.292i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01734646540\)
\(L(\frac12)\) \(\approx\) \(0.01734646540\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - 2.23iT \)
23 \( 1 + (6.71 - 21.9i)T \)
good7 \( 1 + 8.25iT - 49T^{2} \)
11 \( 1 + 1.54iT - 121T^{2} \)
13 \( 1 + 9.82T + 169T^{2} \)
17 \( 1 - 30.3iT - 289T^{2} \)
19 \( 1 + 35.7iT - 361T^{2} \)
29 \( 1 - 26.2T + 841T^{2} \)
31 \( 1 - 20.1T + 961T^{2} \)
37 \( 1 + 29.5iT - 1.36e3T^{2} \)
41 \( 1 - 4.35T + 1.68e3T^{2} \)
43 \( 1 - 14.5iT - 1.84e3T^{2} \)
47 \( 1 + 21.5T + 2.20e3T^{2} \)
53 \( 1 + 82.8iT - 2.80e3T^{2} \)
59 \( 1 - 13.9T + 3.48e3T^{2} \)
61 \( 1 + 91.8iT - 3.72e3T^{2} \)
67 \( 1 - 88.3iT - 4.48e3T^{2} \)
71 \( 1 + 77.9T + 5.04e3T^{2} \)
73 \( 1 + 101.T + 5.32e3T^{2} \)
79 \( 1 - 41.8iT - 6.24e3T^{2} \)
83 \( 1 - 83.7iT - 6.88e3T^{2} \)
89 \( 1 + 62.4iT - 7.92e3T^{2} \)
97 \( 1 + 12.9iT - 9.40e3T^{2} \)
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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

−8.440959628156892119081979589647, −7.80829089858874182671055339308, −7.01374419316543047322533105673, −6.62049557349875267734171152621, −5.66120320397616538018826051269, −4.69963184065987774687661708182, −4.04319329973811614233651509703, −3.22231859815527088429401919499, −2.25916510817695209940301275147, −1.12185073795207084661396279810, 0.00374694577612055147857258701, 1.32325042662709994029319004240, 2.44582030491156376689742636675, 2.99172279933416605400369238460, 4.32199834494641699187314786167, 4.91572027723210058629359961479, 5.68250699421017878312847248307, 6.31461483655738693114054534857, 7.28127393115447295446529662534, 7.989846137359238214148873875435

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