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Properties

Label 1-4647-4647.4646-r1-0-0
Degree $1$
Conductor $4647$
Sign $1$
Analytic cond. $499.389$
Root an. cond. $499.389$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 19-s − 20-s − 22-s − 23-s + 25-s − 26-s + 28-s + 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 38-s − 40-s − 41-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 19-s − 20-s − 22-s − 23-s + 25-s − 26-s + 28-s + 29-s − 31-s + 32-s − 34-s − 35-s − 37-s − 38-s − 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4647 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4647 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4647\)    =    \(3 \cdot 1549\)
Sign: $1$
Analytic conductor: \(499.389\)
Root analytic conductor: \(499.389\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4647} (4646, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 4647,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.567089821\)
\(L(\frac12)\) \(\approx\) \(2.567089821\)
\(L(1)\) \(\approx\) \(1.474733394\)
\(L(1)\) \(\approx\) \(1.474733394\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
1549 \( 1 \)
good2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−18.02097778279358415421405422628, −17.21833571563276562525131485500, −16.55089668305371870530487414216, −15.590344672248479302651577794133, −15.42939787037666303127267528147, −14.68908601541856478817966371919, −14.069331690782442616019721784442, −13.34295763856813986191776842011, −12.44716824339163538318021412448, −12.12778836816583515907883556304, −11.390637873272463099470240740740, −10.60919306006935031754556995158, −10.38976371599087520529147403690, −8.86772938882632948358686241170, −8.23283413020380713698641066487, −7.49492832032334557409511387028, −7.11987135326240137071842882807, −6.08880273573768458501503213006, −5.24184149046845519461724568560, −4.549147419314789395785398529650, −4.25739578857661321950344684017, −3.2150676506910252690656470146, −2.36062016751776301897340888621, −1.828340930080732296973388627431, −0.440178101685580357037065119947, 0.440178101685580357037065119947, 1.828340930080732296973388627431, 2.36062016751776301897340888621, 3.2150676506910252690656470146, 4.25739578857661321950344684017, 4.549147419314789395785398529650, 5.24184149046845519461724568560, 6.08880273573768458501503213006, 7.11987135326240137071842882807, 7.49492832032334557409511387028, 8.23283413020380713698641066487, 8.86772938882632948358686241170, 10.38976371599087520529147403690, 10.60919306006935031754556995158, 11.390637873272463099470240740740, 12.12778836816583515907883556304, 12.44716824339163538318021412448, 13.34295763856813986191776842011, 14.069331690782442616019721784442, 14.68908601541856478817966371919, 15.42939787037666303127267528147, 15.590344672248479302651577794133, 16.55089668305371870530487414216, 17.21833571563276562525131485500, 18.02097778279358415421405422628

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