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Properties

Label 1-547-547.28-r1-0-0
Degree $1$
Conductor $547$
Sign $-0.577 - 0.816i$
Analytic cond. $58.7833$
Root an. cond. $58.7833$
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  + (0.354 + 0.935i)2-s − 3-s + (−0.748 + 0.663i)4-s + (−0.885 + 0.464i)5-s + (−0.354 − 0.935i)6-s + (0.354 − 0.935i)7-s + (−0.885 − 0.464i)8-s + 9-s + (−0.748 − 0.663i)10-s + (−0.970 + 0.239i)11-s + (0.748 − 0.663i)12-s + 13-s + 14-s + (0.885 − 0.464i)15-s + (0.120 − 0.992i)16-s + (0.970 + 0.239i)17-s + ⋯
L(s)  = 1  + (0.354 + 0.935i)2-s − 3-s + (−0.748 + 0.663i)4-s + (−0.885 + 0.464i)5-s + (−0.354 − 0.935i)6-s + (0.354 − 0.935i)7-s + (−0.885 − 0.464i)8-s + 9-s + (−0.748 − 0.663i)10-s + (−0.970 + 0.239i)11-s + (0.748 − 0.663i)12-s + 13-s + 14-s + (0.885 − 0.464i)15-s + (0.120 − 0.992i)16-s + (0.970 + 0.239i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(58.7833\)
Root analytic conductor: \(58.7833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (1:\ ),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2126069489 + 0.4109725394i\)
\(L(\frac12)\) \(\approx\) \(-0.2126069489 + 0.4109725394i\)
\(L(1)\) \(\approx\) \(0.5551553879 + 0.4169832495i\)
\(L(1)\) \(\approx\) \(0.5551553879 + 0.4169832495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.354 + 0.935i)T \)
3 \( 1 - T \)
5 \( 1 + (-0.885 + 0.464i)T \)
7 \( 1 + (0.354 - 0.935i)T \)
11 \( 1 + (-0.970 + 0.239i)T \)
13 \( 1 + T \)
17 \( 1 + (0.970 + 0.239i)T \)
19 \( 1 + (0.120 + 0.992i)T \)
23 \( 1 + (-0.568 + 0.822i)T \)
29 \( 1 + (0.568 + 0.822i)T \)
31 \( 1 + (-0.120 + 0.992i)T \)
37 \( 1 + (-0.568 - 0.822i)T \)
41 \( 1 - T \)
43 \( 1 + (0.748 - 0.663i)T \)
47 \( 1 + (0.568 + 0.822i)T \)
53 \( 1 + (0.885 - 0.464i)T \)
59 \( 1 + (-0.120 + 0.992i)T \)
61 \( 1 + (0.354 + 0.935i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (-0.120 - 0.992i)T \)
73 \( 1 + (0.885 + 0.464i)T \)
79 \( 1 + (0.748 - 0.663i)T \)
83 \( 1 + (-0.885 + 0.464i)T \)
89 \( 1 + (0.354 - 0.935i)T \)
97 \( 1 + (-0.970 - 0.239i)T \)
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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

−22.6329196675405363928231887003, −21.75772526731644348547023955613, −20.98662457155006827386567052532, −20.41415579624659284671323466833, −19.030082010538397116252033947689, −18.611018027309600674665116407195, −17.909453366699143930691311230337, −16.69936885794721495426595178944, −15.64273124608366641515806618850, −15.273119127343390940235041514668, −13.76288524213165954560852817608, −12.86873005107648974079259160761, −12.13053943837606495490021836965, −11.52330000483948660862318452656, −10.858652814624533837534376346793, −9.85624747788270669673955076435, −8.68158829834531888398188588136, −7.89012132828541656408383871960, −6.28488577087105618675109984278, −5.358133899949848449397327259505, −4.75955754109394516430809807828, −3.69249944965039231883993555268, −2.45794067219854998971101596406, −1.06307603602382330357529331417, −0.15864908946391923280290618112, 1.15909432390533009140954393808, 3.452443779019951549268363893001, 4.063002842714494758341928025858, 5.152002414857285448537961811377, 5.95063205605878435434899715132, 7.11934890218266565141611727539, 7.56414035741764720357645127745, 8.46426970015545552226462023580, 10.18644466176392243085900626857, 10.69124220637430623920001744219, 11.87501479851741362391941626821, 12.53865913746167033497345015769, 13.614586227888814687087031892593, 14.4319346335243867831915434488, 15.51587269524557751180177126422, 16.1139544052995467012903357181, 16.71343229809778266457388975300, 17.88145624876186123948141340762, 18.2412286536155057723791137199, 19.26549877955802657776689454050, 20.71696183793098666452963077318, 21.34438264882861748636802479662, 22.508684589138012325283485477653, 23.157589248919556133388877776429, 23.60659242423814378446672047823

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