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Properties

Label 1-252-252.83-r1-0-0
Degree $1$
Conductor $252$
Sign $0.642 + 0.766i$
Analytic cond. $27.0811$
Root an. cond. $27.0811$
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  + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + 17-s + 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + 37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s − 53-s + 55-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + 17-s + 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + 37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)47-s − 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(27.0811\)
Root analytic conductor: \(27.0811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 252,\ (1:\ ),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.480086521 + 0.6901756796i\)
\(L(\frac12)\) \(\approx\) \(1.480086521 + 0.6901756796i\)
\(L(1)\) \(\approx\) \(1.033998219 + 0.1657419464i\)
\(L(1)\) \(\approx\) \(1.033998219 + 0.1657419464i\)

Euler product

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

−25.63773372814071554343610270992, −24.67854870923453552436600875957, −23.736849494462215422005379810634, −23.14473161581770401837288826632, −21.97958697949751069572453640561, −20.68005103204805613626227853612, −20.451363020596059903763283516890, −19.1396855003431497710946186885, −18.331544639534929639123967976183, −17.12119459423698867921076095632, −16.27211577073849139361872933207, −15.518028911273590787519264457882, −14.33211723927893636889228505908, −13.2671712421355312487799375958, −12.25955061538853320094514797315, −11.58829145254868960072789261092, −10.16690633403022728487410864594, −9.234621339431881432943121456168, −8.12327000235470804977367086064, −7.26770196451346934590789314351, −5.80273663414171062844313565361, −4.70428668362548930345766017071, −3.748898509142854817555213786473, −2.08665297668443450188056742711, −0.66913441859890203678422779921, 1.00902112903575942944947588110, 2.95130030518335197185953278806, 3.53413563213795278984493638362, 5.246196069779862567980022773531, 6.22187168447764165788327289192, 7.55794125200758817526452824367, 8.18232577702235877101476268223, 9.69173424107810420198163782963, 10.70591830838880919157371332679, 11.44631937894428746033878019950, 12.594909051472839162978750541667, 13.78405033624565219503436196758, 14.57060131154072672843021328106, 15.72751048235121027792125436882, 16.28840129758039107405566476667, 17.84328664592536056717421489023, 18.4210716456491971290320173698, 19.38770169100170502320420558, 20.28880127936821929385993009153, 21.45582948153370185147046351490, 22.2344186131216417912241151114, 23.26567781122369902396812222985, 23.8237661058980743546459672584, 25.16394566369824001285108219871, 25.89330310022770871160765057812

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