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Properties

Label 1-621-621.11-r0-0-0
Degree $1$
Conductor $621$
Sign $0.942 + 0.334i$
Analytic cond. $2.88391$
Root an. cond. $2.88391$
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  + (−0.967 − 0.251i)2-s + (0.873 + 0.486i)4-s + (0.991 + 0.126i)5-s + (−0.472 + 0.881i)7-s + (−0.723 − 0.690i)8-s + (−0.928 − 0.371i)10-s + (0.902 + 0.429i)11-s + (−0.999 − 0.0317i)13-s + (0.678 − 0.734i)14-s + (0.527 + 0.849i)16-s + (−0.327 − 0.945i)17-s + (0.327 − 0.945i)19-s + (0.805 + 0.592i)20-s + (−0.766 − 0.642i)22-s + (0.967 + 0.251i)25-s + (0.959 + 0.281i)26-s + ⋯
L(s)  = 1  + (−0.967 − 0.251i)2-s + (0.873 + 0.486i)4-s + (0.991 + 0.126i)5-s + (−0.472 + 0.881i)7-s + (−0.723 − 0.690i)8-s + (−0.928 − 0.371i)10-s + (0.902 + 0.429i)11-s + (−0.999 − 0.0317i)13-s + (0.678 − 0.734i)14-s + (0.527 + 0.849i)16-s + (−0.327 − 0.945i)17-s + (0.327 − 0.945i)19-s + (0.805 + 0.592i)20-s + (−0.766 − 0.642i)22-s + (0.967 + 0.251i)25-s + (0.959 + 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $0.942 + 0.334i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ 0.942 + 0.334i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.045881711 + 0.1798370093i\)
\(L(\frac12)\) \(\approx\) \(1.045881711 + 0.1798370093i\)
\(L(1)\) \(\approx\) \(0.8467176636 + 0.03688056762i\)
\(L(1)\) \(\approx\) \(0.8467176636 + 0.03688056762i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.967 - 0.251i)T \)
5 \( 1 + (0.991 + 0.126i)T \)
7 \( 1 + (-0.472 + 0.881i)T \)
11 \( 1 + (0.902 + 0.429i)T \)
13 \( 1 + (-0.999 - 0.0317i)T \)
17 \( 1 + (-0.327 - 0.945i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (0.857 + 0.513i)T \)
31 \( 1 + (0.805 - 0.592i)T \)
37 \( 1 + (0.888 - 0.458i)T \)
41 \( 1 + (0.386 + 0.922i)T \)
43 \( 1 + (0.916 - 0.400i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (-0.527 + 0.849i)T \)
61 \( 1 + (-0.997 - 0.0634i)T \)
67 \( 1 + (0.823 + 0.567i)T \)
71 \( 1 + (-0.580 - 0.814i)T \)
73 \( 1 + (-0.327 + 0.945i)T \)
79 \( 1 + (-0.950 + 0.312i)T \)
83 \( 1 + (-0.386 + 0.922i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (-0.296 + 0.954i)T \)
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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

−23.014827313524279871275739847412, −22.0253120825833786649737034411, −21.19539666330151993579160193012, −20.25677396263715501032485385829, −19.55358920355707015006700461575, −18.90554794453029754194600164334, −17.690787081153087240092441161414, −17.149700402628359161348081058460, −16.68649986766276125758519162331, −15.70246276411323911044571063514, −14.4405064364839483346727431195, −14.03859361867694209102965114478, −12.78951540886478814124335142491, −11.85159522127823866731342457869, −10.63769815078845666582694635094, −10.05324676818559782397773694793, −9.37376700931056332259998441779, −8.42558624848173497526855565569, −7.37713378953803212463593073872, −6.43625374575658654167771735808, −5.901097198412331968620765826094, −4.48508119255699719225276814411, −3.09655588653165289751059219809, −1.89620814931174699865701711533, −0.901152733753635987295933588787, 1.08539744960320413947647488822, 2.49830893316767135356322484097, 2.73982829503667008491034459817, 4.56119751365378274919795432294, 5.828184710946999739683575460391, 6.66691609473599748287138086460, 7.421924240949240011368878823973, 8.86597325798327194767884407928, 9.393566450490367196791014637231, 9.90327107371001151658567334609, 11.07684189941498063938604806224, 12.004391278507242227116702473127, 12.645679597721817259179703942519, 13.80045738434474037464971713138, 14.85069351642139564386234390876, 15.67371813696142506755247981655, 16.63015091209407524979489961250, 17.4590666811643414649654778851, 17.98615350456764315207570303407, 18.84387223354972092158887814723, 19.71382892379852714842302126135, 20.34311659183575228591402130007, 21.52636996952158134679715348236, 21.93499497688220683908242698590, 22.70941255614013077858095667304

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