Properties

Label 2-4600-5.4-c1-0-25
Degree $2$
Conductor $4600$
Sign $0.447 - 0.894i$
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  + i·7-s + 3·9-s − 6·11-s + 2i·13-s − 3i·17-s + 6·19-s i·23-s − 3·29-s − 3·31-s + i·37-s + 9·41-s + 8i·43-s + 4i·47-s + 6·49-s i·53-s + ⋯
L(s)  = 1  + 0.377i·7-s + 9-s − 1.80·11-s + 0.554i·13-s − 0.727i·17-s + 1.37·19-s − 0.208i·23-s − 0.557·29-s − 0.538·31-s + 0.164i·37-s + 1.40·41-s + 1.21i·43-s + 0.583i·47-s + 0.857·49-s − 0.137i·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $0.447 - 0.894i$
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: \((2,\ 4600,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.633242703\)
\(L(\frac12)\) \(\approx\) \(1.633242703\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−8.332500620763886995163268329317, −7.50760568985551690456235254807, −7.32455680407539074097898174167, −6.23306749822150298562826978274, −5.35202882263727020787012978441, −4.90217461302165383951306066785, −3.96414004902763441188966460948, −2.91103777254490309407546115060, −2.23419283449759969299113854842, −0.992050436900914754116202054484, 0.52597335362696331861488673489, 1.73684536868741167577433724392, 2.75557830537893932966884538548, 3.63685219384694221354548819077, 4.43718168230733038584775330490, 5.42755694932790245043144064116, 5.70561057142759781889338646595, 7.05327383951325524737312396510, 7.46721978945700448306441662723, 7.974684030038371362661499655364

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