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Properties

Label 1-1224-1224.11-r1-0-0
Degree $1$
Conductor $1224$
Sign $0.529 + 0.848i$
Analytic cond. $131.537$
Root an. cond. $131.537$
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  + (−0.991 − 0.130i)5-s + (0.991 − 0.130i)7-s + (0.130 + 0.991i)11-s + (−0.866 − 0.5i)13-s + (0.707 + 0.707i)19-s + (0.793 − 0.608i)23-s + (0.965 + 0.258i)25-s + (−0.608 + 0.793i)29-s + (0.130 − 0.991i)31-s − 35-s + (0.923 − 0.382i)37-s + (−0.608 − 0.793i)41-s + (−0.965 − 0.258i)43-s + (−0.866 + 0.5i)47-s + (0.965 − 0.258i)49-s + ⋯
L(s)  = 1  + (−0.991 − 0.130i)5-s + (0.991 − 0.130i)7-s + (0.130 + 0.991i)11-s + (−0.866 − 0.5i)13-s + (0.707 + 0.707i)19-s + (0.793 − 0.608i)23-s + (0.965 + 0.258i)25-s + (−0.608 + 0.793i)29-s + (0.130 − 0.991i)31-s − 35-s + (0.923 − 0.382i)37-s + (−0.608 − 0.793i)41-s + (−0.965 − 0.258i)43-s + (−0.866 + 0.5i)47-s + (0.965 − 0.258i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.529 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.529 + 0.848i$
Analytic conductor: \(131.537\)
Root analytic conductor: \(131.537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1224,\ (1:\ ),\ 0.529 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.400232927 + 0.7762355934i\)
\(L(\frac12)\) \(\approx\) \(1.400232927 + 0.7762355934i\)
\(L(1)\) \(\approx\) \(0.9819002721 + 0.06627185174i\)
\(L(1)\) \(\approx\) \(0.9819002721 + 0.06627185174i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.991 - 0.130i)T \)
7 \( 1 + (0.991 - 0.130i)T \)
11 \( 1 + (0.130 + 0.991i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.793 - 0.608i)T \)
29 \( 1 + (-0.608 + 0.793i)T \)
31 \( 1 + (0.130 - 0.991i)T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (-0.608 - 0.793i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (0.258 + 0.965i)T \)
61 \( 1 + (-0.991 + 0.130i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (0.130 + 0.991i)T \)
83 \( 1 + (0.258 - 0.965i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.608 + 0.793i)T \)
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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

−20.8531091399792939606384901107, −19.88373896879704842357259249143, −19.37739796002024421139579329525, −18.583479012199586457696917080475, −17.835703312023619110275308644407, −16.86961926084918533886817389749, −16.282828204222355443448167664764, −15.23472320573253664463040717091, −14.81799683428510729991309075471, −13.91724929263927094099352203535, −13.10955618429753231343831969713, −11.84307023881115320248776883220, −11.56712260194992482797558813942, −10.91348984318144116332495376817, −9.731975451332227904588257396570, −8.791391330767482647486346140370, −8.07381666198372720349578626553, −7.363692265824536058180536547241, −6.52024473402716194987995899852, −5.196619819735219430122162781636, −4.700595772427543708348087798537, −3.584905562805819961757022031089, −2.78448541455966360996589849062, −1.49743361144528138469410924123, −0.425452973590479585167056569671, 0.80877381200860423548922218641, 1.87908348803539333716354887547, 3.00766130052686592972266603917, 4.10890501960302329555977220979, 4.7661723734706933454805889819, 5.521257617180404324079292094338, 7.01072721873411847753550855755, 7.530017939377329082741768569896, 8.191244038934263335481930172639, 9.163889795994986822678965838234, 10.13376852655463728373868496429, 10.95972644268379942157006571159, 11.79801909977664160937209211301, 12.33042875075663846103075125170, 13.16930156717185111859794382486, 14.43574503904655265270236822195, 14.87697394708376982731631280950, 15.46747397087771678371615258197, 16.62424675867912409446564430118, 17.10338510367408197634509431001, 18.12871314263042712925965791529, 18.65220019270409023361871309958, 19.764980870159239727314225219387, 20.27655280020704936729065546307, 20.807452219430449002941811603893

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