Properties

Label 1-547-547.161-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.920 - 0.390i$
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  + (−0.154 − 0.987i)2-s + (0.623 + 0.781i)3-s + (−0.952 + 0.305i)4-s + (0.449 − 0.893i)5-s + (0.675 − 0.736i)6-s + (−0.289 − 0.957i)7-s + (0.449 + 0.893i)8-s + (−0.222 + 0.974i)9-s + (−0.952 − 0.305i)10-s + (0.120 − 0.992i)11-s + (−0.832 − 0.553i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (0.978 − 0.205i)15-s + (0.813 − 0.582i)16-s + (−0.994 − 0.103i)17-s + ⋯
L(s)  = 1  + (−0.154 − 0.987i)2-s + (0.623 + 0.781i)3-s + (−0.952 + 0.305i)4-s + (0.449 − 0.893i)5-s + (0.675 − 0.736i)6-s + (−0.289 − 0.957i)7-s + (0.449 + 0.893i)8-s + (−0.222 + 0.974i)9-s + (−0.952 − 0.305i)10-s + (0.120 − 0.992i)11-s + (−0.832 − 0.553i)12-s + (−0.900 + 0.433i)13-s + (−0.900 + 0.433i)14-s + (0.978 − 0.205i)15-s + (0.813 − 0.582i)16-s + (−0.994 − 0.103i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1978081414 - 0.9730220556i\)
\(L(\frac12)\) \(\approx\) \(0.1978081414 - 0.9730220556i\)
\(L(1)\) \(\approx\) \(0.7861881080 - 0.5499429503i\)
\(L(1)\) \(\approx\) \(0.7861881080 - 0.5499429503i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.154 - 0.987i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.449 - 0.893i)T \)
7 \( 1 + (-0.289 - 0.957i)T \)
11 \( 1 + (0.120 - 0.992i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.994 - 0.103i)T \)
19 \( 1 + (0.386 - 0.922i)T \)
23 \( 1 + (-0.999 - 0.0345i)T \)
29 \( 1 + (-0.650 - 0.759i)T \)
31 \( 1 + (0.0517 - 0.998i)T \)
37 \( 1 + (0.256 + 0.966i)T \)
41 \( 1 + T \)
43 \( 1 + (0.509 + 0.860i)T \)
47 \( 1 + (0.885 - 0.464i)T \)
53 \( 1 + (0.978 - 0.205i)T \)
59 \( 1 + (-0.748 - 0.663i)T \)
61 \( 1 + (-0.928 + 0.370i)T \)
67 \( 1 + (-0.832 - 0.553i)T \)
71 \( 1 + (-0.479 - 0.877i)T \)
73 \( 1 + (0.978 + 0.205i)T \)
79 \( 1 + (0.509 + 0.860i)T \)
83 \( 1 + (-0.970 - 0.239i)T \)
89 \( 1 + (-0.928 - 0.370i)T \)
97 \( 1 + (-0.539 - 0.842i)T \)
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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

−24.02304928995678965926808496437, −22.85745879715033264139441748967, −22.397877512135404198114837607301, −21.51342157211020752762810123428, −20.063933577337978274308854974805, −19.36059605859343974214857550350, −18.38174112217972215608085332991, −18.00324550893105545456717995974, −17.320336038685939088311488069690, −15.89332942733881918921880684581, −15.06514156345600740833099929319, −14.55804831681455669562307019342, −13.81275072742832875651191692509, −12.705517340491489482434182700133, −12.171251067943267823901709950363, −10.4632957517154454972407563942, −9.52015508693225739759084203595, −8.91888805131140137158163894546, −7.667658199568774215114073310694, −7.15488481514407355494939037181, −6.208355696291071454694765854250, −5.48643752443813447693869866351, −3.96398335737862796878651461589, −2.67127940242790211265150798940, −1.79682645683935385266008467322, 0.48269003698392783370685887473, 1.98477608426846997422892839382, 2.94108891447927661346005258713, 4.21573082330296820417888214278, 4.530377588740333727649655303582, 5.82952198909073581752886600282, 7.56384679319832579684161929008, 8.49598546892100941833602387039, 9.4059477625668083123800609214, 9.76764259812805373767485523178, 10.85757068800345683247892269171, 11.60794963964634486629750465763, 12.90554058039323073207466270595, 13.71231352352181638871068437757, 13.96519028383703344238559175172, 15.408802428056127566432998057665, 16.65074047075402132771425586771, 16.87906446560949918551662487972, 17.9992836526867116689287419351, 19.36240107172288377584729938562, 19.79398417861734953945505965832, 20.44863245490451643813000023155, 21.21603577078755850850318724990, 21.977297904411033140387292658200, 22.51189978071008142017515432667

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