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Properties

Label 1-4600-4600.1517-r0-0-0
Degree $1$
Conductor $4600$
Sign $0.368 - 0.929i$
Analytic cond. $21.3623$
Root an. cond. $21.3623$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 + 0.309i)3-s − i·7-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.587 − 0.809i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (0.309 + 0.951i)31-s + (−0.951 + 0.309i)33-s + (0.587 − 0.809i)37-s + (0.809 − 0.587i)39-s + (−0.809 − 0.587i)41-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)3-s − i·7-s + (0.809 + 0.587i)9-s + (−0.809 + 0.587i)11-s + (0.587 − 0.809i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (0.587 + 0.809i)27-s + (0.309 − 0.951i)29-s + (0.309 + 0.951i)31-s + (−0.951 + 0.309i)33-s + (0.587 − 0.809i)37-s + (0.809 − 0.587i)39-s + (−0.809 − 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $0.368 - 0.929i$
Analytic conductor: \(21.3623\)
Root analytic conductor: \(21.3623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (1517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4600,\ (0:\ ),\ 0.368 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.996230128 - 1.356636594i\)
\(L(\frac12)\) \(\approx\) \(1.996230128 - 1.356636594i\)
\(L(1)\) \(\approx\) \(1.452833334 - 0.2049434405i\)
\(L(1)\) \(\approx\) \(1.452833334 - 0.2049434405i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.951 + 0.309i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.587 - 0.809i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.951 + 0.309i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.44209622663960276364266046663, −18.11949969156637887488791202994, −16.74367056077274495401017033425, −16.379439567943882037923661683417, −15.400954719446516631319295173333, −15.09279750261329590498777481040, −14.19216997954605448916288532355, −13.78266431800491437614535004576, −12.90919585621324818979502472594, −12.41021820511828968662064535440, −11.70781953592558505451942195527, −10.84990911323985081036063163717, −9.982999379558284508000849878936, −9.37565919575694016696236643434, −8.59375320472400012577690210971, −8.13803999858094415190169879529, −7.57140780592247758454821164538, −6.38540268872941418593141238309, −6.040377774863946317840457616441, −5.054861194428947413553576766396, −4.14619843486465698315088178191, −3.25237239170784121955300962666, −2.79311984840023466807219912055, −1.8387777687016805890155936297, −1.21271381135486303404282604738, 0.562081895688356361409801949867, 1.55282801914120810583259536286, 2.53970903814122904481352312759, 3.17700590054626179873554975851, 3.9059280462152943268500510169, 4.67835188995016170882236587142, 5.29663385749408242351747846282, 6.40676356207902484765812338995, 7.35719755931997983544414506631, 7.68335787452768469453587549644, 8.401839512856658270083199098991, 9.213569397379698673178054174510, 9.9771858623955183765659741006, 10.50472679960958280879302169047, 10.95603641512757421078431217496, 12.1990194082567611814450610866, 12.83891178788077380637841752830, 13.60274067771479460562674485316, 13.86328366667506973741389542496, 14.76620705156081437936786242573, 15.46957594301959230749953532003, 15.84324825991709512097809942050, 16.67864541957055311958123500850, 17.466200632171882026881607515154, 18.07552572968244573659281023445

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