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Properties

Label 1-55-55.43-r0-0-0
Degree $1$
Conductor $55$
Sign $-0.850 + 0.525i$
Analytic cond. $0.255418$
Root an. cond. $0.255418$
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  + i·2-s + i·3-s − 4-s − 6-s + i·7-s − i·8-s − 9-s − i·12-s − i·13-s − 14-s + 16-s + i·17-s − i·18-s + 19-s − 21-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s + i·7-s − i·8-s − 9-s − i·12-s − i·13-s − 14-s + 16-s + i·17-s − i·18-s + 19-s − 21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(0.255418\)
Root analytic conductor: \(0.255418\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 55,\ (0:\ ),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2095296766 + 0.7375752672i\)
\(L(\frac12)\) \(\approx\) \(0.2095296766 + 0.7375752672i\)
\(L(1)\) \(\approx\) \(0.5681825932 + 0.6908970148i\)
\(L(1)\) \(\approx\) \(0.5681825932 + 0.6908970148i\)

Euler product

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

−32.36846222489207584606431003901, −31.082281997136659420455691185834, −30.40228607668609348983391073773, −29.267205040598442276655835442490, −28.694028133055617448315206884096, −27.074604754124605231722921176582, −26.11274123694921689375997661899, −24.44524607677033870636521768993, −23.35837913542121270306685369852, −22.47438114365105007497242594099, −20.81606794903912392965766668972, −19.913957582672861860996791713581, −18.848247921416292540248810854615, −17.84589802006902908395438349915, −16.64768212483750881771031802450, −14.14459172543860594237408538551, −13.601241601754318067842414513151, −12.19129976380949201954467889363, −11.23110802001590907634234554970, −9.7198889901411094722380200637, −8.1835189412554803394645940004, −6.7426346968167754797254702577, −4.692444821709162981566826113627, −2.8983068119797527576009841846, −1.17155576578275287654596552465, 3.36821907198822085139335372618, 5.05253762399085146339272978849, 5.99327372809511249394729296399, 8.0202951658370410192258206025, 9.1138940445649639357805213867, 10.31416767897367231854698175471, 12.16537747871155698957330730347, 13.822996443558026762192561296946, 15.232324529844928647522678750276, 15.63748650676019008009904308135, 17.05887107232046072157090689955, 18.11319170454102123632588394468, 19.670082160581298652304251505741, 21.36936236511018472333035383780, 22.18706704960063304768598865595, 23.24629128720234115597461017040, 24.769621705110224692389199204, 25.6249728337038806296669379924, 26.75242517794643683342436564489, 27.73930261508734232438839379529, 28.57715824334067897918165744472, 30.6956928537032805792928413518, 31.78150812078579489952822386590, 32.549007131244852016923870742324, 33.55974636240407206389437111767

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