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Properties

Label 1-547-547.96-r0-0-0
Degree $1$
Conductor $547$
Sign $0.205 - 0.978i$
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  + (0.278 − 0.960i)2-s + 3-s + (−0.845 − 0.534i)4-s + (−0.200 + 0.979i)5-s + (0.278 − 0.960i)6-s + (0.692 − 0.721i)7-s + (−0.748 + 0.663i)8-s + 9-s + (0.885 + 0.464i)10-s + (−0.632 − 0.774i)11-s + (−0.845 − 0.534i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.200 + 0.979i)15-s + (0.428 + 0.903i)16-s + (0.987 + 0.160i)17-s + ⋯
L(s)  = 1  + (0.278 − 0.960i)2-s + 3-s + (−0.845 − 0.534i)4-s + (−0.200 + 0.979i)5-s + (0.278 − 0.960i)6-s + (0.692 − 0.721i)7-s + (−0.748 + 0.663i)8-s + 9-s + (0.885 + 0.464i)10-s + (−0.632 − 0.774i)11-s + (−0.845 − 0.534i)12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.200 + 0.979i)15-s + (0.428 + 0.903i)16-s + (0.987 + 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (96, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.205 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.646775372 - 1.336652340i\)
\(L(\frac12)\) \(\approx\) \(1.646775372 - 1.336652340i\)
\(L(1)\) \(\approx\) \(1.408503822 - 0.7278412955i\)
\(L(1)\) \(\approx\) \(1.408503822 - 0.7278412955i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.278 - 0.960i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.200 + 0.979i)T \)
7 \( 1 + (0.692 - 0.721i)T \)
11 \( 1 + (-0.632 - 0.774i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.987 + 0.160i)T \)
19 \( 1 + (0.428 - 0.903i)T \)
23 \( 1 + (0.799 - 0.600i)T \)
29 \( 1 + (0.120 - 0.992i)T \)
31 \( 1 + (0.568 - 0.822i)T \)
37 \( 1 + (0.799 + 0.600i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.0402 + 0.999i)T \)
47 \( 1 + (-0.919 + 0.391i)T \)
53 \( 1 + (0.948 - 0.316i)T \)
59 \( 1 + (0.428 + 0.903i)T \)
61 \( 1 + (0.278 - 0.960i)T \)
67 \( 1 + (-0.0402 + 0.999i)T \)
71 \( 1 + (-0.996 + 0.0804i)T \)
73 \( 1 + (-0.200 - 0.979i)T \)
79 \( 1 + (0.885 - 0.464i)T \)
83 \( 1 + (0.948 - 0.316i)T \)
89 \( 1 + (-0.970 - 0.239i)T \)
97 \( 1 + (-0.632 + 0.774i)T \)
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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

−23.75069954513027348740888207154, −23.0226101587691282131847695597, −21.773255941418910511502392964939, −20.994877230291211329704254085882, −20.49045376641714407266474979680, −19.3520579802127226278490297667, −18.33866084474977553262721877801, −17.69312741777267014773390172423, −16.59486987668541220738473860581, −15.72064281825011731696108647761, −15.098659285950187984850479878530, −14.435173048516827654271955785775, −13.44341970966045936553493186586, −12.50661418070442207077236403326, −12.167914911326329675140689176153, −10.173040469978322679590912854, −9.3299166339987299138251023601, −8.45226247285751638089126592225, −7.869532986957355102930578469614, −7.18956367825407529057526262593, −5.28619755177237805102182935012, −5.20070973856318734479103744051, −3.89284897923258939849395287602, −2.81305949196567063334489588475, −1.36823996186351384063698495158, 1.09748351037103238056382532483, 2.4447836221934270343373319986, 3.037477153672984010205321217124, 4.06805757816487038336424787644, 4.8935946120514235747595099239, 6.45462039060697712758751503837, 7.632280429886573981182377514725, 8.30315797621149231247759261840, 9.5738109485776554294660134261, 10.224868286198184903795250013669, 11.1534579168218215942116383837, 11.80530092517257100075822417227, 13.310600845282896391201580019176, 13.651795936284712365049130723971, 14.689769862469277679296627843476, 14.91752290810495765300596067956, 16.392151121253445333893411204555, 17.67417799431403447586688258466, 18.60710760827177498513749278857, 19.10550411291674351195270173622, 19.80016341742618517691302201041, 20.86056288700602122328408424113, 21.28614344790355086060277419068, 22.09027815737244136193846576934, 23.21141587400649855097758569251

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