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Properties

Label 1-547-547.305-r0-0-0
Degree $1$
Conductor $547$
Sign $0.917 - 0.396i$
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  + (0.0517 − 0.998i)2-s + (−0.222 + 0.974i)3-s + (−0.994 − 0.103i)4-s + (−0.154 − 0.987i)5-s + (0.962 + 0.272i)6-s + (0.813 + 0.582i)7-s + (−0.154 + 0.987i)8-s + (−0.900 − 0.433i)9-s + (−0.994 + 0.103i)10-s + (0.885 + 0.464i)11-s + (0.322 − 0.946i)12-s + (0.623 − 0.781i)13-s + (0.623 − 0.781i)14-s + (0.997 + 0.0689i)15-s + (0.978 + 0.205i)16-s + (−0.999 + 0.0345i)17-s + ⋯
L(s)  = 1  + (0.0517 − 0.998i)2-s + (−0.222 + 0.974i)3-s + (−0.994 − 0.103i)4-s + (−0.154 − 0.987i)5-s + (0.962 + 0.272i)6-s + (0.813 + 0.582i)7-s + (−0.154 + 0.987i)8-s + (−0.900 − 0.433i)9-s + (−0.994 + 0.103i)10-s + (0.885 + 0.464i)11-s + (0.322 − 0.946i)12-s + (0.623 − 0.781i)13-s + (0.623 − 0.781i)14-s + (0.997 + 0.0689i)15-s + (0.978 + 0.205i)16-s + (−0.999 + 0.0345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.917 - 0.396i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.917 - 0.396i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191401724 - 0.2464242505i\)
\(L(\frac12)\) \(\approx\) \(1.191401724 - 0.2464242505i\)
\(L(1)\) \(\approx\) \(0.9719994604 - 0.2325137510i\)
\(L(1)\) \(\approx\) \(0.9719994604 - 0.2325137510i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.0517 - 0.998i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (-0.154 - 0.987i)T \)
7 \( 1 + (0.813 + 0.582i)T \)
11 \( 1 + (0.885 + 0.464i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (-0.999 + 0.0345i)T \)
19 \( 1 + (-0.792 + 0.609i)T \)
23 \( 1 + (0.509 + 0.860i)T \)
29 \( 1 + (0.725 + 0.688i)T \)
31 \( 1 + (-0.0172 - 0.999i)T \)
37 \( 1 + (-0.0862 + 0.996i)T \)
41 \( 1 + T \)
43 \( 1 + (0.940 - 0.338i)T \)
47 \( 1 + (-0.354 - 0.935i)T \)
53 \( 1 + (0.997 + 0.0689i)T \)
59 \( 1 + (-0.970 + 0.239i)T \)
61 \( 1 + (0.386 + 0.922i)T \)
67 \( 1 + (0.322 - 0.946i)T \)
71 \( 1 + (0.770 + 0.636i)T \)
73 \( 1 + (0.997 - 0.0689i)T \)
79 \( 1 + (0.940 - 0.338i)T \)
83 \( 1 + (0.568 + 0.822i)T \)
89 \( 1 + (0.386 - 0.922i)T \)
97 \( 1 + (0.188 - 0.982i)T \)
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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

−23.40732693896089457896998356841, −23.00910550302630751202673755073, −22.070429780405190818232416849855, −21.21013496706831635448214107891, −19.58445317506795670920876803783, −19.14509160785274835768494342730, −18.09625708617636112186986834918, −17.6367209399226577534384482574, −16.8658108192699662126189183014, −15.85070066752306562425633752844, −14.67288152382237757416220535794, −14.134169043501247470387438158331, −13.583043127770231937465737812215, −12.44639500924745116756629988079, −11.23202454001985850387715932884, −10.80688114195621561851285575276, −9.07941432312159095112909235446, −8.35138128770226653890855975526, −7.38125640672270525599275390289, −6.57723072899287605412148521056, −6.262033080231294567498070945101, −4.725228846560144764155679693641, −3.8280909797860858788844608655, −2.31481893866008486337778560147, −0.8714341729873561857748677155, 1.04881680022485380012861933089, 2.220926975566614125013203022491, 3.66163065919810356493146310218, 4.39252168792340563841544290554, 5.13247173814362902654277997722, 6.01096571746369696623625232550, 8.0995684635312773660894669271, 8.81830094148325633512877472012, 9.36634379812412883246675490034, 10.48586782371537853855987028904, 11.30975096378560870022862498061, 11.970022058880973926701316339289, 12.799059046531061644752424254649, 13.86380727776913715744956342822, 14.96006603782522264695600338608, 15.494833115499255413398407971061, 16.838005091933851423342863690453, 17.43044828303602311816033242648, 18.18961531294631053810398055874, 19.5644333159364626781760097139, 20.14639338309310631010407543935, 20.92735613199109556565051608207, 21.392607594515367644141822450315, 22.31819441115105135290785331231, 23.037895452916261899774377467187

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