Properties

Label 1-547-547.492-r0-0-0
Degree $1$
Conductor $547$
Sign $0.683 + 0.730i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (0.962 − 0.272i)2-s + (0.623 + 0.781i)3-s + (0.851 − 0.524i)4-s + (0.675 + 0.736i)5-s + (0.813 + 0.582i)6-s + (−0.985 + 0.171i)7-s + (0.675 − 0.736i)8-s + (−0.222 + 0.974i)9-s + (0.851 + 0.524i)10-s + (0.885 − 0.464i)11-s + (0.940 + 0.338i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.154 + 0.987i)15-s + (0.449 − 0.893i)16-s + (−0.650 + 0.759i)17-s + ⋯
L(s)  = 1  + (0.962 − 0.272i)2-s + (0.623 + 0.781i)3-s + (0.851 − 0.524i)4-s + (0.675 + 0.736i)5-s + (0.813 + 0.582i)6-s + (−0.985 + 0.171i)7-s + (0.675 − 0.736i)8-s + (−0.222 + 0.974i)9-s + (0.851 + 0.524i)10-s + (0.885 − 0.464i)11-s + (0.940 + 0.338i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (−0.154 + 0.987i)15-s + (0.449 − 0.893i)16-s + (−0.650 + 0.759i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.683 + 0.730i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (492, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.683 + 0.730i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.928606613 + 1.270191953i\)
\(L(\frac12)\) \(\approx\) \(2.928606613 + 1.270191953i\)
\(L(1)\) \(\approx\) \(2.221415100 + 0.5009023755i\)
\(L(1)\) \(\approx\) \(2.221415100 + 0.5009023755i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.962 - 0.272i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.675 + 0.736i)T \)
7 \( 1 + (-0.985 + 0.171i)T \)
11 \( 1 + (0.885 - 0.464i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.650 + 0.759i)T \)
19 \( 1 + (0.770 - 0.636i)T \)
23 \( 1 + (0.725 + 0.688i)T \)
29 \( 1 + (0.990 + 0.137i)T \)
31 \( 1 + (-0.418 - 0.908i)T \)
37 \( 1 + (-0.832 - 0.553i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.700 - 0.713i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
53 \( 1 + (-0.154 + 0.987i)T \)
59 \( 1 + (-0.970 - 0.239i)T \)
61 \( 1 + (-0.479 - 0.877i)T \)
67 \( 1 + (0.940 + 0.338i)T \)
71 \( 1 + (-0.0172 - 0.999i)T \)
73 \( 1 + (-0.154 - 0.987i)T \)
79 \( 1 + (-0.700 - 0.713i)T \)
83 \( 1 + (0.568 - 0.822i)T \)
89 \( 1 + (-0.479 + 0.877i)T \)
97 \( 1 + (-0.999 - 0.0345i)T \)
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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

−23.134634927923158706558469649344, −22.64598862473746998331338192108, −21.69499785925891037640365540035, −20.65915159664963844004164117113, −19.98176811722384934185935522094, −19.55279782517643063427355216299, −18.09876393475985800933006365012, −17.23746493589478586989287712601, −16.47599759190301669083746095318, −15.51692334591573588405502779098, −14.42814426906532542236633397887, −13.894817558356167796157838986588, −12.9906302205106859228991308621, −12.47766277285144240706492973182, −11.796479364561575541816685084, −10.10579622136766928593743088138, −9.26853855942352742896485035941, −8.28184209196596691533670787038, −7.00445766951269459802646997090, −6.65761889559555116863122250256, −5.494809295158334624447487336792, −4.47354099833043005255491124033, −3.23853949927707416767285947920, −2.4330020818995861860743492510, −1.23386616551137491633098719674, 1.86546592585596888876925578275, 2.865261968494064274931522813232, 3.45619553333870833800310531487, 4.52258374776101073039803856784, 5.64253982135630480626468899685, 6.51637705972808876738366869018, 7.359702292263001693338344464586, 9.1711738916002152206698878499, 9.59005663553949939215609171223, 10.60319574262273264578457454534, 11.324728401418421118674593452816, 12.50605941975520914312888909778, 13.57840253401867541740271023716, 14.01142522737540427339277372669, 14.93173393553557561022412210484, 15.53047682153351585938624425503, 16.499549116080309633888401725203, 17.37511573754529799076183767953, 19.0473229802549875638069510118, 19.41237682826316456379088877917, 20.18027510369277007288309570144, 21.355152616624748368493963049612, 22.02447767954573917849702202926, 22.1353116200734036610034353121, 23.16063500983813237483531591472

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