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Properties

Label 1-252-252.103-r0-0-0
Degree $1$
Conductor $252$
Sign $0.888 - 0.458i$
Analytic cond. $1.17028$
Root an. cond. $1.17028$
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 + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)29-s + 31-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)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 + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)29-s + 31-s + (−0.5 − 0.866i)37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(1.17028\)
Root analytic conductor: \(1.17028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 252,\ (0:\ ),\ 0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.336446772 - 0.3240892587i\)
\(L(\frac12)\) \(\approx\) \(1.336446772 - 0.3240892587i\)
\(L(1)\) \(\approx\) \(1.184105311 - 0.1546444630i\)
\(L(1)\) \(\approx\) \(1.184105311 - 0.1546444630i\)

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 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
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

−25.951201507298816090590060726245, −25.3146846311435150920321418604, −24.33571219587144904686933130446, −23.139870629606345191259989921078, −22.47165201938192140483258556791, −21.460655911329166728876213964961, −20.78977610066953982403642501815, −19.27683400377744418436423016248, −18.86734622046501826735070505523, −17.62398017488957617581555364736, −16.97118947362055054584759454060, −15.64814832193687911568329895935, −14.76134307855943158836437776636, −13.86531551170746258546735087223, −12.974452749623976574786225468707, −11.65765485448507749762250451845, −10.698086963066583896323879949233, −9.92996000345857923226429391178, −8.62063278488458386881401111724, −7.60058398142545892853572795025, −6.229157771206709094349773103494, −5.701988024778586026646542460873, −3.86374314682587663306779174512, −2.955832376913812120626815265187, −1.447695769309955605703543587984, 1.19778630852133713647622857337, 2.4584023457374545703023362580, 4.19413491057448229031176576678, 5.00659121843417770666475282268, 6.30929166315584393612572077848, 7.33200888277462136762368875649, 8.84737987552285701098446739942, 9.30761850970821141615066131155, 10.56042600259884159557304446284, 11.82566775934404303020693622015, 12.63892465234265973137646366609, 13.65267026672375883135230007099, 14.54047805196797256307802466275, 15.78487792210065653716759729042, 16.68447510971830710415908654084, 17.44884805281358328779460225697, 18.46265444534904806813914109523, 19.57914093260888273352442026890, 20.55811939483902558669214232331, 21.162963217488513984752608499688, 22.25176208520653077252817969732, 23.26277365148177203877832775528, 24.17102467217769724852056924118, 25.079904178267448992603468538929, 25.74617997179451392215026621168

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