Properties

Label 2-4600-5.4-c1-0-19
Degree $2$
Conductor $4600$
Sign $-0.894 + 0.447i$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
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  + 3.08i·3-s + 0.555i·7-s − 6.50·9-s + 4.65·11-s + 5.02i·13-s − 1.32i·17-s + 0.196·19-s − 1.71·21-s + i·23-s − 10.8i·27-s + 0.812·29-s − 2.11·31-s + 14.3i·33-s + 5.64i·37-s − 15.4·39-s + ⋯
L(s)  = 1  + 1.78i·3-s + 0.209i·7-s − 2.16·9-s + 1.40·11-s + 1.39i·13-s − 0.322i·17-s + 0.0450·19-s − 0.373·21-s + 0.208i·23-s − 2.08i·27-s + 0.150·29-s − 0.379·31-s + 2.49i·33-s + 0.928i·37-s − 2.47·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(36.7311\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.458793017\)
\(L(\frac12)\) \(\approx\) \(1.458793017\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 - iT \)
good3 \( 1 - 3.08iT - 3T^{2} \)
7 \( 1 - 0.555iT - 7T^{2} \)
11 \( 1 - 4.65T + 11T^{2} \)
13 \( 1 - 5.02iT - 13T^{2} \)
17 \( 1 + 1.32iT - 17T^{2} \)
19 \( 1 - 0.196T + 19T^{2} \)
29 \( 1 - 0.812T + 29T^{2} \)
31 \( 1 + 2.11T + 31T^{2} \)
37 \( 1 - 5.64iT - 37T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 - 1.66iT - 43T^{2} \)
47 \( 1 - 9.89iT - 47T^{2} \)
53 \( 1 - 2.23iT - 53T^{2} \)
59 \( 1 + 2.43T + 59T^{2} \)
61 \( 1 + 5.71T + 61T^{2} \)
67 \( 1 + 6.91iT - 67T^{2} \)
71 \( 1 - 0.0120T + 71T^{2} \)
73 \( 1 - 15.2iT - 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 5.64iT - 83T^{2} \)
89 \( 1 + 9.06T + 89T^{2} \)
97 \( 1 - 14.0iT - 97T^{2} \)
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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

−8.986044574324703029379170836818, −8.391074547385065686818043891607, −7.19402786542196088827306978407, −6.40652709546045054265054612492, −5.72010371572184008887307984519, −4.74447618325584683929463090631, −4.30879632026447153912540167576, −3.64223857408394492642024258566, −2.79974815355376405602585664428, −1.50519136670070217635022270491, 0.41737490862760289317709817446, 1.28556164745002018135063204824, 2.11698395623133677032634983056, 3.12378295902883809445142020515, 3.93277425213611337285165369806, 5.23953550150774112962439739873, 5.95416898387189371132413670886, 6.53391672541393475795493889273, 7.23968630317741979924948044070, 7.69916961197109021041009148702

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