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Properties

Label 1-621-621.265-r0-0-0
Degree $1$
Conductor $621$
Sign $0.0299 - 0.999i$
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  + (−0.823 − 0.567i)2-s + (0.356 + 0.934i)4-s + (0.296 − 0.954i)5-s + (−0.204 − 0.978i)7-s + (0.235 − 0.971i)8-s + (−0.786 + 0.618i)10-s + (−0.266 + 0.963i)11-s + (0.950 − 0.312i)13-s + (−0.386 + 0.922i)14-s + (−0.745 + 0.666i)16-s + (0.981 + 0.189i)17-s + (0.981 − 0.189i)19-s + (0.997 − 0.0634i)20-s + (0.766 − 0.642i)22-s + (−0.823 − 0.567i)25-s + (−0.959 − 0.281i)26-s + ⋯
L(s)  = 1  + (−0.823 − 0.567i)2-s + (0.356 + 0.934i)4-s + (0.296 − 0.954i)5-s + (−0.204 − 0.978i)7-s + (0.235 − 0.971i)8-s + (−0.786 + 0.618i)10-s + (−0.266 + 0.963i)11-s + (0.950 − 0.312i)13-s + (−0.386 + 0.922i)14-s + (−0.745 + 0.666i)16-s + (0.981 + 0.189i)17-s + (0.981 − 0.189i)19-s + (0.997 − 0.0634i)20-s + (0.766 − 0.642i)22-s + (−0.823 − 0.567i)25-s + (−0.959 − 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0299 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0299 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $0.0299 - 0.999i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ 0.0299 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7282248079 - 0.7503357168i\)
\(L(\frac12)\) \(\approx\) \(0.7282248079 - 0.7503357168i\)
\(L(1)\) \(\approx\) \(0.7486802233 - 0.3802994493i\)
\(L(1)\) \(\approx\) \(0.7486802233 - 0.3802994493i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.823 - 0.567i)T \)
5 \( 1 + (0.296 - 0.954i)T \)
7 \( 1 + (-0.204 - 0.978i)T \)
11 \( 1 + (-0.266 + 0.963i)T \)
13 \( 1 + (0.950 - 0.312i)T \)
17 \( 1 + (0.981 + 0.189i)T \)
19 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (0.630 + 0.776i)T \)
31 \( 1 + (0.997 + 0.0634i)T \)
37 \( 1 + (0.0475 - 0.998i)T \)
41 \( 1 + (0.678 + 0.734i)T \)
43 \( 1 + (-0.553 - 0.832i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (-0.745 - 0.666i)T \)
61 \( 1 + (0.805 - 0.592i)T \)
67 \( 1 + (0.967 + 0.251i)T \)
71 \( 1 + (-0.995 + 0.0950i)T \)
73 \( 1 + (0.981 - 0.189i)T \)
79 \( 1 + (-0.999 - 0.0317i)T \)
83 \( 1 + (0.678 - 0.734i)T \)
89 \( 1 + (0.723 - 0.690i)T \)
97 \( 1 + (0.991 + 0.126i)T \)
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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

−23.1848732838389970493010683704, −22.58894667660010301529348016712, −21.44355756538745119579350989304, −20.859736218582329481231598659590, −19.43413962270113287445151520309, −18.819632772118502331442466312185, −18.37546868126514219149233058760, −17.610535527673387746030654894993, −16.400576669500906694179221881645, −15.85808667369111229416717852295, −15.02925003029860297892658102574, −14.12549062233874231346221333887, −13.487718500920779556007938298622, −11.814355271791669932062024945778, −11.28490780057455636129875310611, −10.187849186997236305083296135420, −9.57165881491438192933757483031, −8.49312101631802830314311294891, −7.8342131085083392208163882827, −6.570292510680017653520140830403, −6.031641298047852613970062601369, −5.234270017757690729650014867059, −3.3537803831244381099333085661, −2.52399092026237623302366126559, −1.18725828209532836737494598574, 0.86277479103090848405456101947, 1.58330140772445588100428587693, 3.04612504345205472359197346486, 4.03216095353885116091544700240, 5.060079853952636328832066810829, 6.42212196425397980829349601895, 7.53198486738491576608122204675, 8.162993200948245811661071660451, 9.2538370492734127443388055720, 9.96086036522187245188811300901, 10.63509909780350820267489520378, 11.77282694053005650937192008439, 12.63104969493311403351551105008, 13.23786656615543739642598779340, 14.19132002280344048704237537399, 15.79543010006095447309936153301, 16.22028474897485543994122039819, 17.20841279392486534720044628815, 17.71402506128806608402729832387, 18.60445860142916261724657292468, 19.76426894351494755718388985041, 20.26264443033415975317039676354, 20.84654407861713174898078193770, 21.601654327578531925769235605990, 22.88868641521374185790831643106

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