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Properties

Label 1-621-621.25-r0-0-0
Degree $1$
Conductor $621$
Sign $-0.850 - 0.525i$
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  + (−0.0792 − 0.996i)2-s + (−0.987 + 0.158i)4-s + (0.678 − 0.734i)5-s + (−0.745 − 0.666i)7-s + (0.235 + 0.971i)8-s + (−0.786 − 0.618i)10-s + (0.967 + 0.251i)11-s + (−0.204 − 0.978i)13-s + (−0.605 + 0.795i)14-s + (0.950 − 0.312i)16-s + (0.981 − 0.189i)17-s + (0.981 + 0.189i)19-s + (−0.553 + 0.832i)20-s + (0.173 − 0.984i)22-s + (−0.0792 − 0.996i)25-s + (−0.959 + 0.281i)26-s + ⋯
L(s)  = 1  + (−0.0792 − 0.996i)2-s + (−0.987 + 0.158i)4-s + (0.678 − 0.734i)5-s + (−0.745 − 0.666i)7-s + (0.235 + 0.971i)8-s + (−0.786 − 0.618i)10-s + (0.967 + 0.251i)11-s + (−0.204 − 0.978i)13-s + (−0.605 + 0.795i)14-s + (0.950 − 0.312i)16-s + (0.981 − 0.189i)17-s + (0.981 + 0.189i)19-s + (−0.553 + 0.832i)20-s + (0.173 − 0.984i)22-s + (−0.0792 − 0.996i)25-s + (−0.959 + 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3514044781 - 1.237158070i\)
\(L(\frac12)\) \(\approx\) \(0.3514044781 - 1.237158070i\)
\(L(1)\) \(\approx\) \(0.7535717828 - 0.7135902610i\)
\(L(1)\) \(\approx\) \(0.7535717828 - 0.7135902610i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.0792 - 0.996i)T \)
5 \( 1 + (0.678 - 0.734i)T \)
7 \( 1 + (-0.745 - 0.666i)T \)
11 \( 1 + (0.967 + 0.251i)T \)
13 \( 1 + (-0.204 - 0.978i)T \)
17 \( 1 + (0.981 - 0.189i)T \)
19 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + (0.356 + 0.934i)T \)
31 \( 1 + (-0.553 - 0.832i)T \)
37 \( 1 + (0.0475 + 0.998i)T \)
41 \( 1 + (-0.975 - 0.220i)T \)
43 \( 1 + (-0.444 - 0.895i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (-0.142 - 0.989i)T \)
59 \( 1 + (0.950 + 0.312i)T \)
61 \( 1 + (-0.916 + 0.400i)T \)
67 \( 1 + (-0.701 - 0.712i)T \)
71 \( 1 + (-0.995 - 0.0950i)T \)
73 \( 1 + (0.981 + 0.189i)T \)
79 \( 1 + (0.472 - 0.881i)T \)
83 \( 1 + (-0.975 + 0.220i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (-0.386 + 0.922i)T \)
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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

−23.28163062961098537495212896553, −22.54727555410891255373549513763, −21.863496735990850356446585099354, −21.349138223808918052075136223227, −19.66468766143088330718301903612, −18.95330455825335434571337417589, −18.37016836121965982536965938271, −17.428665555454769656326409292172, −16.6426304274653993228114009216, −15.93512041747220748252620478525, −14.90535465430599675085153481642, −14.22678348012723179871536432965, −13.64700795157202886662727549314, −12.5025558121830827077839277940, −11.585913977058079251432289125657, −10.191337573966397053873436902498, −9.4567197601391287183588349248, −8.93433636008294975775572169122, −7.57445012145137598905159697704, −6.69405949267991183930861179270, −6.098914618914273128833672499294, −5.27625029606340009948863869471, −3.89796774722292360233906421186, −2.93761024527724873714924083136, −1.43238656356501171368804985853, 0.756306102475433860857568526171, 1.614495607568286581275346408396, 3.02899737558179022506036936967, 3.79027904344053675244581699953, 4.97179954240199071477964209814, 5.7627722562391852025462733330, 7.122774507336851552646652147917, 8.254611690211609474860481249658, 9.283903278461850614955934544603, 9.89270018903988759109754585903, 10.49083674758559681132265799675, 11.89068509629351709928839145817, 12.4019643976105650831241528127, 13.35155592168376837307552425608, 13.8716723372687696491516610024, 14.9013455277710352676340843036, 16.41968822249464270204395830424, 16.92253897402236585629126199257, 17.73330271813430485808451470064, 18.60949479015312239790112136936, 19.64477637011353379483888599059, 20.309869315600339391909723626584, 20.66248386896020391080494913630, 21.968937643032606976879341120513, 22.347187865402774028762321658184

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