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Properties

Label 1-547-547.287-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.897 - 0.440i$
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  + (−0.614 + 0.788i)2-s + (−0.900 + 0.433i)3-s + (−0.244 − 0.969i)4-s + (−0.806 − 0.591i)5-s + (0.211 − 0.977i)6-s + (−0.819 − 0.572i)7-s + (0.915 + 0.402i)8-s + (0.623 − 0.781i)9-s + (0.962 − 0.272i)10-s + (0.692 − 0.721i)11-s + (0.641 + 0.767i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (0.983 + 0.183i)15-s + (−0.880 + 0.474i)16-s + (−0.577 − 0.816i)17-s + ⋯
L(s)  = 1  + (−0.614 + 0.788i)2-s + (−0.900 + 0.433i)3-s + (−0.244 − 0.969i)4-s + (−0.806 − 0.591i)5-s + (0.211 − 0.977i)6-s + (−0.819 − 0.572i)7-s + (0.915 + 0.402i)8-s + (0.623 − 0.781i)9-s + (0.962 − 0.272i)10-s + (0.692 − 0.721i)11-s + (0.641 + 0.767i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (0.983 + 0.183i)15-s + (−0.880 + 0.474i)16-s + (−0.577 − 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.897 - 0.440i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.897 - 0.440i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03889192980 - 0.1677341949i\)
\(L(\frac12)\) \(\approx\) \(0.03889192980 - 0.1677341949i\)
\(L(1)\) \(\approx\) \(0.4020874241 + 0.0008461071809i\)
\(L(1)\) \(\approx\) \(0.4020874241 + 0.0008461071809i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.614 + 0.788i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (-0.806 - 0.591i)T \)
7 \( 1 + (-0.819 - 0.572i)T \)
11 \( 1 + (0.692 - 0.721i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.577 - 0.816i)T \)
19 \( 1 + (-0.177 - 0.984i)T \)
23 \( 1 + (0.785 + 0.618i)T \)
29 \( 1 + (0.997 - 0.0689i)T \)
31 \( 1 + (-0.539 - 0.842i)T \)
37 \( 1 + (0.973 + 0.228i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.991 - 0.126i)T \)
47 \( 1 + (-0.996 - 0.0804i)T \)
53 \( 1 + (-0.332 - 0.942i)T \)
59 \( 1 + (-0.919 - 0.391i)T \)
61 \( 1 + (-0.999 + 0.0115i)T \)
67 \( 1 + (0.343 - 0.939i)T \)
71 \( 1 + (0.968 - 0.250i)T \)
73 \( 1 + (0.983 - 0.183i)T \)
79 \( 1 + (0.386 + 0.922i)T \)
83 \( 1 + (-0.845 + 0.534i)T \)
89 \( 1 + (0.509 - 0.860i)T \)
97 \( 1 + (-0.857 + 0.514i)T \)
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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

−23.34616395644193838423161695959, −22.817700926054431843863547024355, −22.023395005553079403014546997215, −21.531510473501255606164336896501, −19.942938571078031305599765027596, −19.453791185747093182880015049085, −18.74971208028010955866672866115, −18.11405573969647303806700976159, −17.05730308424594128511197165243, −16.48760803221149302497084509094, −15.47715990442952622401451519248, −14.36907089119402018264750233363, −12.92959850385165408188680098590, −12.26091611673488897754629036169, −11.86438687570456919152890630819, −10.81267021245631883781995483750, −10.12447101804330575490392737945, −9.08241056454748421004571824961, −7.98937639054101096513053270005, −6.90502297691872020484987719260, −6.48017530847011639690580147263, −4.74612011105316626028119023829, −3.85700460015435990563694152940, −2.62045316039063227072632996663, −1.56632987217288436444699293325, 0.166747703370262045026692615291, 0.96598729068972391674041466263, 3.294283992993292394208339571732, 4.534011317354079726293899737, 5.138446519482033843655835671710, 6.40420944548859999733967334692, 6.993756299644608837356425458993, 8.04751031429066368270546474003, 9.25814976911999017840893692933, 9.71576605339690295538612140813, 10.972964787141855729691538239014, 11.50556236581230385049324795917, 12.77917113696656526864564243162, 13.61854632909432329654172752769, 15.07925854766518390400946568995, 15.56054248304198452280667214786, 16.52704428927056434100913531336, 16.81428267773285632050803968633, 17.67910656709471689271894041036, 18.72229158748699417867388188915, 19.82705371375595900537326322718, 19.98322192905552297454427132058, 21.58538408696821954670757038056, 22.56199988110524621178866344480, 23.06481095839046864495980012728

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