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Properties

Label 1-55-55.37-r1-0-0
Degree $1$
Conductor $55$
Sign $0.998 - 0.0509i$
Analytic cond. $5.91057$
Root an. cond. $5.91057$
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  + (−0.951 + 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + i·12-s + (0.951 − 0.309i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.951 + 0.309i)17-s + (0.587 + 0.809i)18-s + (0.809 + 0.587i)19-s + 21-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + i·12-s + (0.951 − 0.309i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.951 + 0.309i)17-s + (0.587 + 0.809i)18-s + (0.809 + 0.587i)19-s + 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.998 - 0.0509i$
Analytic conductor: \(5.91057\)
Root analytic conductor: \(5.91057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (1:\ ),\ 0.998 - 0.0509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7484616253 + 0.01907876892i\)
\(L(\frac12)\) \(\approx\) \(0.7484616253 + 0.01907876892i\)
\(L(1)\) \(\approx\) \(0.6163827331 + 0.08905143846i\)
\(L(1)\) \(\approx\) \(0.6163827331 + 0.08905143846i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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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

−33.24621674652445905135298541573, −31.371503130589660286278833066565, −30.30737192079687353608167226415, −29.240220294305608870146623839687, −28.42830150328469290109385116297, −27.586077880047096001629881712535, −25.87942696241736325185009644285, −25.17179634530296635255112126318, −23.91976923851351165788273766669, −22.498938391494291930039986530616, −21.26154439299629059532543989221, −19.71301543100906993127650241944, −18.71950409935262520412071750276, −18.00434273949785516673475083302, −16.66098512016301513367476001051, −15.669664494511695980610517818492, −13.457015210653557414569908073601, −12.157353286162630709694478351598, −11.335953859769432083167515867470, −9.75424733910661939123376116100, −8.373409640451886274533744488885, −7.0004695767335708138007093085, −5.77085977827002237771890230046, −2.93722983947841912891144227221, −1.1978145755402085723541765587, 0.75035636352011742738407224993, 3.5488411235447611735382960501, 5.5729348910396304893484854153, 6.823234583240222076536071693186, 8.50386718641188739054797759899, 10.00036964853372296332272426493, 10.60855650223915854769834485611, 12.09498402689116765413278702499, 14.18822097166114706041453106269, 15.6970558631187669245989963398, 16.44588239322257358693925769518, 17.43678902622106882327585227717, 18.70186310325976238548615688377, 20.16029939582020531707894780850, 21.03629092509502655414439549169, 22.78397007363319867048723357341, 23.58324936015139494024388294595, 25.233502206334605995320724834519, 26.33566825469337702749443292562, 27.09850521159285965606026794306, 28.25038529388461480086611300741, 29.04150426298800103160872111315, 30.24175240052944136471083630599, 32.37692539609439443280445403310, 32.95517450456803278474476783513

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