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Properties

Label 1-252-252.67-r1-0-0
Degree $1$
Conductor $252$
Sign $-0.400 - 0.916i$
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  + 5-s − 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s − 23-s + 25-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + 5-s − 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s − 23-s + 25-s + (−0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (−0.5 − 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.400 - 0.916i$
Analytic conductor: \(27.0811\)
Root analytic conductor: \(27.0811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 252,\ (1:\ ),\ -0.400 - 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7232579281 - 1.105909586i\)
\(L(\frac12)\) \(\approx\) \(0.7232579281 - 1.105909586i\)
\(L(1)\) \(\approx\) \(1.021425531 - 0.2308952016i\)
\(L(1)\) \(\approx\) \(1.021425531 - 0.2308952016i\)

Euler product

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

−26.32117185691257409647594743550, −25.09699320048238171698480792888, −24.334570005439574598176205187202, −23.392036445713547887764916281127, −22.23818044407413134723899284157, −21.43710428109325304620719658671, −20.76372698592695943867041370823, −19.586246436125593120412338098699, −18.52098277698656006699116278066, −17.74902039016309399582746626356, −16.80825152452723743174195429032, −15.86580620167086942601125685497, −14.641883629097987645378577276183, −13.80215387421080880947649132763, −12.92293970187875612134284147425, −11.87045686145918525775361194291, −10.52066719876684225550003368567, −9.862348935021183614564753480759, −8.733797628891699662669864620920, −7.568462044346576710328694424782, −6.307289711660895530382058658688, −5.441458815060059624921288787137, −4.19346447929159342797457650737, −2.60760509017926336389638239076, −1.61542692321552271865875450331, 0.385701533562361819059990040996, 2.10994337600446301953291861113, 3.04553766168942424031064716981, 4.87096949085885975988827686409, 5.58047045787562809269519905878, 6.85769417170000329177176316369, 7.94364439946902832493878940489, 9.21986002924554366168697482783, 10.07592531963679657941923244606, 10.97203544090098589169915445718, 12.319268610609507769599546177156, 13.31352581974860447868269976607, 13.9601412629241911980867192082, 15.23740641621182069163891757843, 16.06628627209880902375410773214, 17.34436068420998868761397942530, 17.93407616368839729110342912709, 18.82495781752433530154664678260, 20.27887862955715783887345087839, 20.70503389286487291528460498650, 22.05802828481465778736956587150, 22.39306053147840461402649030598, 23.87385268983195201134960979864, 24.53690075616478297499533327214, 25.6072014454123606846292532567

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