Properties

Label 1-547-547.484-r0-0-0
Degree $1$
Conductor $547$
Sign $0.404 - 0.914i$
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  + (−0.0632 + 0.997i)2-s + (−0.900 − 0.433i)3-s + (−0.991 − 0.126i)4-s + (−0.944 − 0.327i)5-s + (0.490 − 0.871i)6-s + (0.233 − 0.972i)7-s + (0.188 − 0.982i)8-s + (0.623 + 0.781i)9-s + (0.386 − 0.922i)10-s + (0.278 + 0.960i)11-s + (0.838 + 0.544i)12-s + (−0.733 + 0.680i)13-s + (0.955 + 0.294i)14-s + (0.709 + 0.705i)15-s + (0.968 + 0.250i)16-s + (−0.131 + 0.991i)17-s + ⋯
L(s)  = 1  + (−0.0632 + 0.997i)2-s + (−0.900 − 0.433i)3-s + (−0.991 − 0.126i)4-s + (−0.944 − 0.327i)5-s + (0.490 − 0.871i)6-s + (0.233 − 0.972i)7-s + (0.188 − 0.982i)8-s + (0.623 + 0.781i)9-s + (0.386 − 0.922i)10-s + (0.278 + 0.960i)11-s + (0.838 + 0.544i)12-s + (−0.733 + 0.680i)13-s + (0.955 + 0.294i)14-s + (0.709 + 0.705i)15-s + (0.968 + 0.250i)16-s + (−0.131 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.404 - 0.914i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.404 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3346440676 - 0.2179669040i\)
\(L(\frac12)\) \(\approx\) \(0.3346440676 - 0.2179669040i\)
\(L(1)\) \(\approx\) \(0.5416669786 + 0.08592647544i\)
\(L(1)\) \(\approx\) \(0.5416669786 + 0.08592647544i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.0632 + 0.997i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.944 - 0.327i)T \)
7 \( 1 + (0.233 - 0.972i)T \)
11 \( 1 + (0.278 + 0.960i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (-0.131 + 0.991i)T \)
19 \( 1 + (0.407 - 0.913i)T \)
23 \( 1 + (0.300 + 0.953i)T \)
29 \( 1 + (-0.289 - 0.957i)T \)
31 \( 1 + (0.322 - 0.946i)T \)
37 \( 1 + (-0.558 - 0.829i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.244 - 0.969i)T \)
47 \( 1 + (0.428 + 0.903i)T \)
53 \( 1 + (-0.965 + 0.261i)T \)
59 \( 1 + (0.799 - 0.600i)T \)
61 \( 1 + (0.740 + 0.671i)T \)
67 \( 1 + (-0.890 + 0.454i)T \)
71 \( 1 + (-0.880 - 0.474i)T \)
73 \( 1 + (0.709 - 0.705i)T \)
79 \( 1 + (0.962 + 0.272i)T \)
83 \( 1 + (-0.0402 - 0.999i)T \)
89 \( 1 + (-0.952 - 0.305i)T \)
97 \( 1 + (-0.998 - 0.0575i)T \)
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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

−23.19225177174511227849005797967, −22.32019803024592604858402887266, −22.14785454443888162519780231210, −21.0826883163495380391976016058, −20.28659282428700810538065645081, −19.27148861296798444398546931089, −18.455423024533771903984645359974, −18.04726160052166536843420498926, −16.77935976495814190489698033388, −16.05845931757881164239757417404, −15.00108704516263611603564987036, −14.27387627946632436262875404347, −12.780712323693196026337853330546, −12.04136464416401822085552152935, −11.57906496179365183798031027817, −10.77740482048683047493049469471, −9.933613602933496298386739368932, −8.85891753463171666563194574849, −8.03233779951066306896717878255, −6.65793637112255131184577671149, −5.34361602719788247297652471942, −4.79672360327339110719506923434, −3.506079067698002074949392946674, −2.823407430439690626142157779449, −1.0828279036518657654665064217, 0.30661099595278183292009841025, 1.62019185408905829176822206052, 3.99092916761171721256670050786, 4.487044149043620736451326855198, 5.40912600748012576043154924125, 6.7300174787010022140503753053, 7.30716935215859151224244396433, 7.84879080703802377090003474494, 9.18397778450666503413231903776, 10.18401750608578336337045742118, 11.27014549180082470683902458319, 12.1485421837464807319370412488, 13.00880891547959630917111136113, 13.83506608986983315960813651776, 15.00632825557239492750517995421, 15.662070540777463827931524458537, 16.66835544654355429701962783480, 17.29886411266931971414320723436, 17.668737104960414705979226852321, 19.1730988332048128141648905469, 19.39554802762652708363732453054, 20.705003265910215565275266356377, 22.06559215157544838545648591540, 22.64466474775834964465522398014, 23.58250743849370796216530535016

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