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Properties

Label 1-547-547.506-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.862 - 0.506i$
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  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + 10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + 10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.862 - 0.506i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.862 - 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2027653888 - 0.7460862663i\)
\(L(\frac12)\) \(\approx\) \(0.2027653888 - 0.7460862663i\)
\(L(1)\) \(\approx\) \(0.7144638209 - 0.4043328436i\)
\(L(1)\) \(\approx\) \(0.7144638209 - 0.4043328436i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−24.0251547603825674602589261176, −23.26450249096694198378168131386, −22.02135326609556695541782769164, −21.17012269789745694264721350851, −19.97700619796204492727729181687, −19.54760512540306737839327146985, −18.79498176865918789226843845326, −17.869489824744159979385502500733, −16.82883389909872499264957901945, −15.93011058277966524344316430875, −15.315733099693478589702619055149, −14.789819629499980356888531369459, −13.53689399395741713425327821222, −12.83708223644029663290389649030, −11.92859096851999684393312261828, −10.25025787973602089637474081389, −9.55449181833386605893932914607, −8.59450872998953605703996741642, −8.33927073937986072144051749426, −7.11714076115873710470447124742, −6.32302419807975228804553774239, −4.748340275883119873941615101204, −4.397947843219483821568120035044, −2.638576034996278539847605602617, −1.58145016674911888226924552560, 0.413339908449348183891137363119, 2.0763321236105626919585706315, 3.197664814788388067428104700898, 3.458549705887457881367530219221, 4.671030345216059223138828737165, 6.5755622651293955747845289436, 7.66933142996702627756210072748, 8.007162177249161754333429502305, 9.20814333194461790019418939353, 10.33792637063404945134454600001, 10.47414903658433330313446359853, 11.771877921454666389946503567194, 12.76649128285635822926473508959, 13.73015966300794406627916651180, 14.13456373006875458678905912624, 15.5383952956194653931468736192, 16.13441503184719503786357711883, 17.4352418434286456941904324536, 18.285447625577764558737545729181, 19.18977776343843499297893098703, 19.52451027156411950796737579942, 20.36315401264507267214862147747, 21.119742759661783919183210549781, 22.08013844610911409741033671387, 22.81775925689326016644281789209

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