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Properties

Label 2-4140-5.4-c1-0-48
Degree $2$
Conductor $4140$
Sign $-0.930 + 0.365i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
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  + (0.817 + 2.08i)5-s − 4.41i·7-s − 2.29·11-s + 6.92i·13-s + 1.51i·17-s − 2.89·19-s − i·23-s + (−3.66 + 3.40i)25-s − 7.68·29-s + 3.85·31-s + (9.18 − 3.60i)35-s − 8.62i·37-s + 6.44·41-s − 3.48i·43-s − 6.19i·47-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)5-s − 1.66i·7-s − 0.691·11-s + 1.92i·13-s + 0.367i·17-s − 0.665·19-s − 0.208i·23-s + (−0.732 + 0.680i)25-s − 1.42·29-s + 0.692·31-s + (1.55 − 0.609i)35-s − 1.41i·37-s + 1.00·41-s − 0.531i·43-s − 0.903i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.930 + 0.365i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.930 + 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1408632663\)
\(L(\frac12)\) \(\approx\) \(0.1408632663\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.817 - 2.08i)T \)
23 \( 1 + iT \)
good7 \( 1 + 4.41iT - 7T^{2} \)
11 \( 1 + 2.29T + 11T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 - 1.51iT - 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
29 \( 1 + 7.68T + 29T^{2} \)
31 \( 1 - 3.85T + 31T^{2} \)
37 \( 1 + 8.62iT - 37T^{2} \)
41 \( 1 - 6.44T + 41T^{2} \)
43 \( 1 + 3.48iT - 43T^{2} \)
47 \( 1 + 6.19iT - 47T^{2} \)
53 \( 1 + 2.17iT - 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 5.11T + 61T^{2} \)
67 \( 1 + 9.94iT - 67T^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + 8.95iT - 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 + 8.04iT - 83T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 - 16.9iT - 97T^{2} \)
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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

−7.69499241258641379340650570775, −7.46398064782765396317679039458, −6.59682937046933942735630838833, −6.21207032618176682312661645246, −4.99599644964996665057914433844, −4.07046847086904874880016375730, −3.69843010454443344190474566200, −2.39802684251788955132224781395, −1.63567043620041701107223041886, −0.03752494361453562399073407931, 1.37556843612408292970145288129, 2.55522435632986980530060015792, 3.00065309657832481704871899596, 4.43648592896756318925616782765, 5.22069480854226531537058276008, 5.70647897518453060074355485386, 6.15876754570665708209286330094, 7.54666560892448083394600224775, 8.194855266025673353664060159691, 8.600406235056080374567764170936

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