Properties

Label 2-4140-23.22-c2-0-52
Degree $2$
Conductor $4140$
Sign $0.0489 + 0.998i$
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 − 2.98i·7-s + 12.9i·11-s − 9.48·13-s − 6.45i·17-s + 20.7i·19-s + (22.9 − 1.12i)23-s − 5.00·25-s − 35.9·29-s − 1.84·31-s − 6.68·35-s − 40.6i·37-s − 29.8·41-s − 6.92i·43-s − 21.6·47-s + ⋯
L(s)  = 1  − 0.447i·5-s − 0.426i·7-s + 1.17i·11-s − 0.729·13-s − 0.379i·17-s + 1.09i·19-s + (0.998 − 0.0489i)23-s − 0.200·25-s − 1.23·29-s − 0.0594·31-s − 0.190·35-s − 1.09i·37-s − 0.727·41-s − 0.160i·43-s − 0.460·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0489 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0489 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.0489 + 0.998i$
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: \((2,\ 4140,\ (\ :1),\ 0.0489 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.364279444\)
\(L(\frac12)\) \(\approx\) \(1.364279444\)
\(L(2)\) 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 \( 1 + (-22.9 + 1.12i)T \)
good7 \( 1 + 2.98iT - 49T^{2} \)
11 \( 1 - 12.9iT - 121T^{2} \)
13 \( 1 + 9.48T + 169T^{2} \)
17 \( 1 + 6.45iT - 289T^{2} \)
19 \( 1 - 20.7iT - 361T^{2} \)
29 \( 1 + 35.9T + 841T^{2} \)
31 \( 1 + 1.84T + 961T^{2} \)
37 \( 1 + 40.6iT - 1.36e3T^{2} \)
41 \( 1 + 29.8T + 1.68e3T^{2} \)
43 \( 1 + 6.92iT - 1.84e3T^{2} \)
47 \( 1 + 21.6T + 2.20e3T^{2} \)
53 \( 1 + 72.8iT - 2.80e3T^{2} \)
59 \( 1 - 74.5T + 3.48e3T^{2} \)
61 \( 1 - 43.9iT - 3.72e3T^{2} \)
67 \( 1 - 46.0iT - 4.48e3T^{2} \)
71 \( 1 - 101.T + 5.04e3T^{2} \)
73 \( 1 + 14.4T + 5.32e3T^{2} \)
79 \( 1 - 56.4iT - 6.24e3T^{2} \)
83 \( 1 + 10.4iT - 6.88e3T^{2} \)
89 \( 1 + 125. iT - 7.92e3T^{2} \)
97 \( 1 - 35.6iT - 9.40e3T^{2} \)
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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

−8.001239576329926355999108440826, −7.21855744596883006927744068397, −6.92272742570820027064252733828, −5.65254067030231978118386107313, −5.11471768324911845327851063790, −4.28566402547885891257878342081, −3.58601012854759796196077223684, −2.36670042273332068122203035274, −1.57396286789166012943510488243, −0.33717163140608155400859491742, 0.860386516249789650511428774109, 2.17736063654087914630559681495, 2.97456651541694965720655964659, 3.66832988154509951502169935348, 4.82382973362574465659251786552, 5.43792456580991452369357314622, 6.27675853113594725525742177572, 6.92507981107290138652226486081, 7.67827346980334633440950008336, 8.479917200515521408497129654783

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