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Properties

Label 1-4600-4600.1021-r0-0-0
Degree $1$
Conductor $4600$
Sign $0.951 + 0.308i$
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  + (0.985 − 0.170i)3-s + (−0.654 − 0.755i)7-s + (0.941 − 0.336i)9-s + (0.362 + 0.931i)11-s + (−0.0855 + 0.996i)13-s + (0.993 − 0.113i)17-s + (0.736 + 0.676i)19-s + (−0.774 − 0.633i)21-s + (0.870 − 0.491i)27-s + (0.736 − 0.676i)29-s + (−0.985 − 0.170i)31-s + (0.516 + 0.856i)33-s + (−0.941 + 0.336i)37-s + (0.0855 + 0.996i)39-s + (0.610 − 0.791i)41-s + ⋯
L(s)  = 1  + (0.985 − 0.170i)3-s + (−0.654 − 0.755i)7-s + (0.941 − 0.336i)9-s + (0.362 + 0.931i)11-s + (−0.0855 + 0.996i)13-s + (0.993 − 0.113i)17-s + (0.736 + 0.676i)19-s + (−0.774 − 0.633i)21-s + (0.870 − 0.491i)27-s + (0.736 − 0.676i)29-s + (−0.985 − 0.170i)31-s + (0.516 + 0.856i)33-s + (−0.941 + 0.336i)37-s + (0.0855 + 0.996i)39-s + (0.610 − 0.791i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $0.951 + 0.308i$
Analytic conductor: \(21.3623\)
Root analytic conductor: \(21.3623\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4600,\ (0:\ ),\ 0.951 + 0.308i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.666191866 + 0.4219788317i\)
\(L(\frac12)\) \(\approx\) \(2.666191866 + 0.4219788317i\)
\(L(1)\) \(\approx\) \(1.532915988 + 0.005940757968i\)
\(L(1)\) \(\approx\) \(1.532915988 + 0.005940757968i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.985 - 0.170i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
11 \( 1 + (0.362 + 0.931i)T \)
13 \( 1 + (-0.0855 + 0.996i)T \)
17 \( 1 + (0.993 - 0.113i)T \)
19 \( 1 + (0.736 + 0.676i)T \)
29 \( 1 + (0.736 - 0.676i)T \)
31 \( 1 + (-0.985 - 0.170i)T \)
37 \( 1 + (-0.941 + 0.336i)T \)
41 \( 1 + (0.610 - 0.791i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.516 + 0.856i)T \)
59 \( 1 + (-0.0855 + 0.996i)T \)
61 \( 1 + (0.466 - 0.884i)T \)
67 \( 1 + (0.254 + 0.967i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (-0.870 + 0.491i)T \)
79 \( 1 + (-0.921 - 0.389i)T \)
83 \( 1 + (0.564 - 0.825i)T \)
89 \( 1 + (0.696 - 0.717i)T \)
97 \( 1 + (-0.564 - 0.825i)T \)
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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

−18.26346663333497736129112100643, −17.72100874963751141083560640824, −16.44254368349931041686809219118, −16.21401367391193001490289326293, −15.44402007384033019171559018002, −14.82718235865032377097560309350, −14.1836273739360851708881499425, −13.543805915435453433189233144060, −12.76397544744128127056132228220, −12.35213231127606597990636946722, −11.36523212649813424019491729416, −10.56990619196307574552392221649, −9.81148475137530948914126983470, −9.27560392882011940944793459805, −8.584912477967605232047876302273, −8.03045299591598459520390401974, −7.22252633896250853276637731708, −6.42017534190705487096428998081, −5.52614870540334169011419679978, −5.00141255971858598911610654164, −3.73232606946893960562009185645, −3.19377033941085144556998767841, −2.8012402647099960951970184340, −1.70732245663599383248235961042, −0.72602344288642199487009918925, 0.997782888110931271648722237950, 1.71895641178097012660944487295, 2.57015026549364263124973829301, 3.49800324788695325895900263802, 3.97249930032441865066879520309, 4.691102716467891238533615438630, 5.82857851278655264599864629211, 6.70721844784780813526072048743, 7.366490971481390556187020102940, 7.65751964236220364319907453036, 8.79490060218760131779771956130, 9.3642158424578305035448984933, 9.98349661115116196355792348075, 10.43083903906534080274674403445, 11.68567310457605410476281807862, 12.2926255912334133311460032297, 12.852910349085500583435652726198, 13.73594035028141011845331627437, 14.2374519600779245062494408716, 14.5821446667631896433495971415, 15.72869059263536772710844420473, 16.0104842912168219804079318154, 16.98822323068130625932858057762, 17.447486180179839102715963606903, 18.571433924543177167885704260356

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