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Properties

Label 1-1100-1100.103-r0-0-0
Degree $1$
Conductor $1100$
Sign $0.345 + 0.938i$
Analytic cond. $5.10837$
Root an. cond. $5.10837$
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)3-s + (−0.587 − 0.809i)7-s + (0.809 + 0.587i)9-s + (0.587 − 0.809i)13-s + (−0.587 + 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.309 + 0.951i)21-s + (−0.587 + 0.809i)23-s + (−0.587 − 0.809i)27-s + (0.809 + 0.587i)29-s − 31-s + (−0.951 − 0.309i)37-s + (−0.809 + 0.587i)39-s + 41-s − i·43-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)3-s + (−0.587 − 0.809i)7-s + (0.809 + 0.587i)9-s + (0.587 − 0.809i)13-s + (−0.587 + 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.309 + 0.951i)21-s + (−0.587 + 0.809i)23-s + (−0.587 − 0.809i)27-s + (0.809 + 0.587i)29-s − 31-s + (−0.951 − 0.309i)37-s + (−0.809 + 0.587i)39-s + 41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.345 + 0.938i$
Analytic conductor: \(5.10837\)
Root analytic conductor: \(5.10837\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1100,\ (0:\ ),\ 0.345 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4424522303 + 0.3084417088i\)
\(L(\frac12)\) \(\approx\) \(0.4424522303 + 0.3084417088i\)
\(L(1)\) \(\approx\) \(0.6569035522 - 0.04010184951i\)
\(L(1)\) \(\approx\) \(0.6569035522 - 0.04010184951i\)

Euler product

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

−21.4680094059340884556391905365, −20.679872082303291528576609853681, −19.620570827810831583496475274534, −18.73193482231733111798110031653, −18.20577293574005832638602954838, −17.37486414441458264749711940980, −16.46128643415535109795592110759, −15.89992889232460091170634710536, −15.32175806100427224892409522006, −14.23683743945126696758755330235, −13.22662309927903880594961136400, −12.4523684187077069030695193451, −11.74704296544916060252021826841, −11.03004266995420029023662681843, −10.19775470447863306842950507948, −9.21910366255190843613170667013, −8.73328307548147558978987162543, −7.2742313766271743111416683288, −6.389284119891600833740363172, −5.96273603460647386353930270781, −4.78659644921783568829281276656, −4.161330823572613201570627676950, −2.903987295730956315020350748871, −1.83270055685553964195089953185, −0.29964290240542130307128761672, 0.99845489554112997704127304093, 2.03378967913977128777880885185, 3.556762168379514825067127307811, 4.19478790441373013649759831426, 5.42005917926334912138577115269, 6.11592381439370210232631622039, 6.8822005285257868112084618307, 7.70446906354337386622619800106, 8.65074725922541535104355630897, 9.89614598886864818518689527205, 10.62217268108259848375173934960, 11.0259212174414398082586310103, 12.274500891945591309052607361015, 12.82168069422220261304223354115, 13.48412408290825414436982331775, 14.43408593637687786420441682, 15.696185499614220582059489741990, 16.050856016969228925056479956107, 17.143102023120041705610899515571, 17.47490329249929885774697598760, 18.37253440077889279247676250601, 19.24376163099587106108180994278, 19.88606751996427069931661850231, 20.80531142508588098431059293045, 21.769366655079277847271243002288

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