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Properties

Label 1-547-547.321-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.990 + 0.138i$
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  + (−0.0172 − 0.999i)2-s + (−0.900 − 0.433i)3-s + (−0.999 + 0.0345i)4-s + (0.0517 − 0.998i)5-s + (−0.418 + 0.908i)6-s + (0.978 − 0.205i)7-s + (0.0517 + 0.998i)8-s + (0.623 + 0.781i)9-s + (−0.999 − 0.0345i)10-s + (−0.354 + 0.935i)11-s + (0.915 + 0.402i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.479 + 0.877i)15-s + (0.997 − 0.0689i)16-s + (0.509 − 0.860i)17-s + ⋯
L(s)  = 1  + (−0.0172 − 0.999i)2-s + (−0.900 − 0.433i)3-s + (−0.999 + 0.0345i)4-s + (0.0517 − 0.998i)5-s + (−0.418 + 0.908i)6-s + (0.978 − 0.205i)7-s + (0.0517 + 0.998i)8-s + (0.623 + 0.781i)9-s + (−0.999 − 0.0345i)10-s + (−0.354 + 0.935i)11-s + (0.915 + 0.402i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.479 + 0.877i)15-s + (0.997 − 0.0689i)16-s + (0.509 − 0.860i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.990 + 0.138i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.990 + 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06147426077 - 0.8830315852i\)
\(L(\frac12)\) \(\approx\) \(-0.06147426077 - 0.8830315852i\)
\(L(1)\) \(\approx\) \(0.4897473067 - 0.6316570674i\)
\(L(1)\) \(\approx\) \(0.4897473067 - 0.6316570674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.0172 - 0.999i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.0517 - 0.998i)T \)
7 \( 1 + (0.978 - 0.205i)T \)
11 \( 1 + (-0.354 + 0.935i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.509 - 0.860i)T \)
19 \( 1 + (0.675 - 0.736i)T \)
23 \( 1 + (0.940 - 0.338i)T \)
29 \( 1 + (-0.700 - 0.713i)T \)
31 \( 1 + (-0.868 + 0.495i)T \)
37 \( 1 + (0.851 - 0.524i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.596 + 0.802i)T \)
47 \( 1 + (0.120 - 0.992i)T \)
53 \( 1 + (-0.479 + 0.877i)T \)
59 \( 1 + (0.568 - 0.822i)T \)
61 \( 1 + (-0.792 - 0.609i)T \)
67 \( 1 + (0.915 + 0.402i)T \)
71 \( 1 + (-0.289 + 0.957i)T \)
73 \( 1 + (-0.479 - 0.877i)T \)
79 \( 1 + (-0.596 + 0.802i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + (-0.792 + 0.609i)T \)
97 \( 1 + (-0.832 + 0.553i)T \)
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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

−23.86433517464229654281197827346, −23.130475719529901465190249778877, −22.15511836784372434206153823046, −21.65205874065450144962795738672, −20.96216920284569169757099442028, −19.0721898110315882265216942429, −18.52870052348177340470540794189, −17.86954156722609581090566641273, −16.898704190593143206499461333045, −16.392821712089136945683109352563, −15.278836519779446602725986172323, −14.67187107591052391324223195022, −13.990024911741361208695984627085, −12.76453580265848047851008228713, −11.54150283495826794039255334291, −10.93773793334163123497164890829, −9.9556721183363104707595281653, −8.98511688905031264896037672378, −7.79920257765417323544775249667, −7.04416928010908124279888148060, −5.92745472913104506951694435606, −5.48393317251241885992076717760, −4.30899926810195553305014145510, −3.36351658540923325450480813488, −1.42132582402325381230428419440, 0.611755063926397053945562119526, 1.47415454206191334537758730041, 2.58353453178418880864141326643, 4.28540639107046482394149912953, 5.09186645125120471663461414833, 5.43703875287914803786809652911, 7.37901937299288309616990347637, 7.96131182516040247229580514511, 9.237128402441988465102737806420, 10.07778523685309983819426139799, 11.10386915800000881914797220718, 11.69298424136816840847713726229, 12.69096351525430476473391943207, 13.01612148572846841094314156941, 14.11347498610881062374385935234, 15.303767368914045133983631330780, 16.5408480702616047571043173887, 17.38551405584263985988705791252, 17.88031587517419852840584039018, 18.571657960823355812547151290762, 19.880280285188649122376528453843, 20.41494049175940302079220317879, 21.1587975341613699583426580231, 22.03514758472860380806875375002, 23.10463010671134069091755248604

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