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Properties

Label 1-547-547.23-r1-0-0
Degree $1$
Conductor $547$
Sign $-0.154 - 0.987i$
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  + (−0.973 − 0.228i)2-s + (0.222 + 0.974i)3-s + (0.895 + 0.444i)4-s + (0.937 + 0.349i)5-s + (0.00575 − 0.999i)6-s + (−0.143 + 0.989i)7-s + (−0.770 − 0.636i)8-s + (−0.900 + 0.433i)9-s + (−0.832 − 0.553i)10-s + (−0.919 − 0.391i)11-s + (−0.233 + 0.972i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.131 + 0.991i)15-s + (0.605 + 0.795i)16-s + (0.980 + 0.194i)17-s + ⋯
L(s)  = 1  + (−0.973 − 0.228i)2-s + (0.222 + 0.974i)3-s + (0.895 + 0.444i)4-s + (0.937 + 0.349i)5-s + (0.00575 − 0.999i)6-s + (−0.143 + 0.989i)7-s + (−0.770 − 0.636i)8-s + (−0.900 + 0.433i)9-s + (−0.832 − 0.553i)10-s + (−0.919 − 0.391i)11-s + (−0.233 + 0.972i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.131 + 0.991i)15-s + (0.605 + 0.795i)16-s + (0.980 + 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.154 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.154 - 0.987i$
Analytic conductor: \(58.7833\)
Root analytic conductor: \(58.7833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (1:\ ),\ -0.154 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08174368765 + 0.09553747814i\)
\(L(\frac12)\) \(\approx\) \(-0.08174368765 + 0.09553747814i\)
\(L(1)\) \(\approx\) \(0.5854959430 + 0.2950333088i\)
\(L(1)\) \(\approx\) \(0.5854959430 + 0.2950333088i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.973 - 0.228i)T \)
3 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (0.937 + 0.349i)T \)
7 \( 1 + (-0.143 + 0.989i)T \)
11 \( 1 + (-0.919 - 0.391i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (0.980 + 0.194i)T \)
19 \( 1 + (-0.890 + 0.454i)T \)
23 \( 1 + (0.109 + 0.994i)T \)
29 \( 1 + (-0.596 - 0.802i)T \)
31 \( 1 + (-0.813 - 0.582i)T \)
37 \( 1 + (-0.529 + 0.848i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.376 - 0.926i)T \)
47 \( 1 + (-0.0402 - 0.999i)T \)
53 \( 1 + (0.924 - 0.381i)T \)
59 \( 1 + (0.200 + 0.979i)T \)
61 \( 1 + (0.778 + 0.627i)T \)
67 \( 1 + (-0.958 + 0.283i)T \)
71 \( 1 + (0.717 - 0.696i)T \)
73 \( 1 + (-0.131 - 0.991i)T \)
79 \( 1 + (-0.990 + 0.137i)T \)
83 \( 1 + (-0.278 + 0.960i)T \)
89 \( 1 + (0.154 + 0.987i)T \)
97 \( 1 + (-0.880 - 0.474i)T \)
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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

−22.92651677082731094346770853742, −21.40585260155415946829654617072, −20.54192687414411529482903238947, −19.986822708320713411815730828241, −19.10532058965006429372910025369, −18.23120004305541916564275162459, −17.5661640681289221555084598879, −16.935836063192951205668998038842, −16.197338785819976577032869345997, −14.62789555169312328846850556351, −14.275795510531668582150374576223, −12.83411422008732902898651782900, −12.62039550868790293099020823574, −11.05369047171720365000704571949, −10.257295995514771733077226253432, −9.4637254636159775123440183907, −8.47077528159410216928146918247, −7.504635255403864687703577218536, −6.97622524969436769013644665139, −5.93265939567093110335590986965, −4.94471765204070617936364283834, −2.92237416199571428852010418592, −2.09243145148577764287502883984, −1.0497819892554063317148756090, −0.04224584502835676983963853769, 2.01981131688701361090363747644, 2.61368461262441551350855539187, 3.6305238905981128090868865488, 5.4354264475244480945539467919, 5.822522012940310997091527948968, 7.31373382997215104147263459772, 8.36240851449816271037978826825, 9.18009690384608560642607342634, 9.934911871275489273502236235932, 10.43503204781540204852336287004, 11.465188560303087617138637322229, 12.43475729670218588499922285185, 13.58340912959076055027834379045, 14.919622737976115585654727919426, 15.21111917521013726476631277168, 16.42935853912549310114884008605, 16.98276932284810294239017856049, 17.890822410072855970853129684269, 18.888119486440856268705390524402, 19.316822881699563798965108177750, 20.71871329919582702190322767475, 21.18094979948071375364248737123, 21.754845726272977613360388190101, 22.521194871812968815348804278420, 24.02088386536273065492269261241

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