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Properties

Label 1-572-572.95-r0-0-0
Degree $1$
Conductor $572$
Sign $-0.450 + 0.892i$
Analytic cond. $2.65635$
Root an. cond. $2.65635$
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  + (0.104 − 0.994i)3-s + (−0.309 + 0.951i)5-s + (0.104 + 0.994i)7-s + (−0.978 − 0.207i)9-s + (0.913 + 0.406i)15-s + (−0.669 − 0.743i)17-s + (−0.913 + 0.406i)19-s + 21-s + (0.5 + 0.866i)23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.913 − 0.406i)29-s + (0.309 + 0.951i)31-s + (−0.978 − 0.207i)35-s + (−0.913 − 0.406i)37-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)3-s + (−0.309 + 0.951i)5-s + (0.104 + 0.994i)7-s + (−0.978 − 0.207i)9-s + (0.913 + 0.406i)15-s + (−0.669 − 0.743i)17-s + (−0.913 + 0.406i)19-s + 21-s + (0.5 + 0.866i)23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.913 − 0.406i)29-s + (0.309 + 0.951i)31-s + (−0.978 − 0.207i)35-s + (−0.913 − 0.406i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.450 + 0.892i$
Analytic conductor: \(2.65635\)
Root analytic conductor: \(2.65635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 572,\ (0:\ ),\ -0.450 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3089124013 + 0.5016468572i\)
\(L(\frac12)\) \(\approx\) \(0.3089124013 + 0.5016468572i\)
\(L(1)\) \(\approx\) \(0.7991525931 + 0.06923507443i\)
\(L(1)\) \(\approx\) \(0.7991525931 + 0.06923507443i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.104 + 0.994i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.913 - 0.406i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−22.90006528212248424102596453109, −22.15681143989798397608339283308, −21.0163072493588459414519247123, −20.64846029315223239461245950851, −19.831792300545969309637186344433, −19.14788318992170076491432708281, −17.53386790520809886767891248919, −16.94641242186963411417622648944, −16.3930500654333512517097974321, −15.37375617893036756586929908244, −14.744625833323820092453597670112, −13.56558387680329712398355767326, −12.90505843199081974021235555911, −11.69683185395240239983010985241, −10.79648710684253211675437383675, −10.15598468352442315155980485848, −8.94725189928775516272850491199, −8.49340199407722588410668069098, −7.33104466180456757086274706642, −6.07209430760307422831789256354, −4.85121845953438839710562339106, −4.30526811856371752048142336259, −3.46858656681350275403311039316, −1.91194337652154869512693511306, −0.28095723489074674145594140394, 1.72448023454585414746963170657, 2.595706183421849656196362372704, 3.47018272811935859245320332615, 5.0600673658882211786190481862, 6.15929356412411345180112441994, 6.8218129062156013644809095399, 7.76884826760062839764602419369, 8.60046406666524785835571253907, 9.582880455204798749752468549400, 10.974035709655660082644827583736, 11.56070061209760776416331222749, 12.39691895974681358751782771576, 13.311085617333427495067699947750, 14.23795012259587631121523204987, 15.00152531083111712717758387256, 15.70901149957712026080014065863, 17.03914413115150096040473943195, 17.99448647455308605947000051957, 18.47570774276371632323402745968, 19.230652004106839593303312724589, 19.863002872766371052537297827914, 21.12213496955114034151367813329, 21.936890550398923766258127352619, 22.91954070647065955356206985505, 23.323916911188380935099541125981

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