↑

Properties

Label 1-547-547.350-r0-0-0
Degree $1$
Conductor $547$
Sign $0.644 - 0.764i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.568 − 0.822i)2-s + 3-s + (−0.354 − 0.935i)4-s + (−0.970 + 0.239i)5-s + (0.568 − 0.822i)6-s + (0.568 + 0.822i)7-s + (−0.970 − 0.239i)8-s + 9-s + (−0.354 + 0.935i)10-s + (0.120 + 0.992i)11-s + (−0.354 − 0.935i)12-s + 13-s + 14-s + (−0.970 + 0.239i)15-s + (−0.748 + 0.663i)16-s + (0.120 − 0.992i)17-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s + 3-s + (−0.354 − 0.935i)4-s + (−0.970 + 0.239i)5-s + (0.568 − 0.822i)6-s + (0.568 + 0.822i)7-s + (−0.970 − 0.239i)8-s + 9-s + (−0.354 + 0.935i)10-s + (0.120 + 0.992i)11-s + (−0.354 − 0.935i)12-s + 13-s + 14-s + (−0.970 + 0.239i)15-s + (−0.748 + 0.663i)16-s + (0.120 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (350, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.644 - 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.158462155 - 1.003122094i\)
\(L(\frac12)\) \(\approx\) \(2.158462155 - 1.003122094i\)
\(L(1)\) \(\approx\) \(1.646952330 - 0.6158669512i\)
\(L(1)\) \(\approx\) \(1.646952330 - 0.6158669512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.568 - 0.822i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.970 + 0.239i)T \)
7 \( 1 + (0.568 + 0.822i)T \)
11 \( 1 + (0.120 + 0.992i)T \)
13 \( 1 + T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (-0.748 - 0.663i)T \)
23 \( 1 + (0.885 - 0.464i)T \)
29 \( 1 + (0.885 + 0.464i)T \)
31 \( 1 + (-0.748 + 0.663i)T \)
37 \( 1 + (0.885 + 0.464i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.354 - 0.935i)T \)
47 \( 1 + (0.885 + 0.464i)T \)
53 \( 1 + (-0.970 + 0.239i)T \)
59 \( 1 + (-0.748 + 0.663i)T \)
61 \( 1 + (0.568 - 0.822i)T \)
67 \( 1 + (-0.354 - 0.935i)T \)
71 \( 1 + (-0.748 - 0.663i)T \)
73 \( 1 + (-0.970 - 0.239i)T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (0.120 - 0.992i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.55317328214089747967734032462, −23.13304621009306075543760995315, −21.64390251439495571918939824954, −21.08079961328525089769414322585, −20.30832810029766212758558597459, −19.31436847219504588864522632791, −18.60921082271322158902361072180, −17.35241688226545831930840527925, −16.44213172223005782947793596426, −15.82679776070281749799542496333, −14.8247601891296791223775310011, −14.37720796067902956139477345751, −13.30650180497394468886889175492, −12.85135817422510534070886397093, −11.51886177510866594503156873720, −10.6765518700083197069237335479, −9.08864551016387296248103476850, −8.228363717681147916948268841893, −7.94878466807783135781182288769, −6.89835935552720295361901536808, −5.7902035436814079790508564598, −4.2477183673321505048261704688, −3.98692714571317639005878053675, −3.02195203945932890399608951285, −1.21107736703977067377168577496, 1.29282410819859019689121777091, 2.48287038075356960338867155325, 3.1613534841443440695361318045, 4.33896029018483477793870998495, 4.87886609336236471060833140884, 6.49993517675181362432777296184, 7.5223543775179512822524208018, 8.72964610754590178539164588519, 9.15319954467605983837147777288, 10.50372249226193530851049648578, 11.25506311550911092674147504548, 12.24393357126164336765819100358, 12.83030456345068551101231064244, 13.96864139814033118475479822852, 14.741772562484724521240548121861, 15.30537380726402694165622437847, 15.97807613038087096084309980911, 17.907055638370015886571633549349, 18.55455817366764305873202017658, 19.19851089399117983415548350487, 20.13483422801995964118533295789, 20.60838060092980743535198782058, 21.44281259735109100245750438163, 22.30666130552438826403720672497, 23.310118545414616995247027604357

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