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Properties

Label 1-621-621.49-r0-0-0
Degree $1$
Conductor $621$
Sign $0.920 + 0.391i$
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  + (0.110 + 0.993i)2-s + (−0.975 + 0.220i)4-s + (−0.745 − 0.666i)5-s + (−0.0792 + 0.996i)7-s + (−0.327 − 0.945i)8-s + (0.580 − 0.814i)10-s + (−0.553 − 0.832i)11-s + (−0.823 + 0.567i)13-s + (−0.999 + 0.0317i)14-s + (0.902 − 0.429i)16-s + (0.0475 − 0.998i)17-s + (0.0475 + 0.998i)19-s + (0.873 + 0.486i)20-s + (0.766 − 0.642i)22-s + (0.110 + 0.993i)25-s + (−0.654 − 0.755i)26-s + ⋯
L(s)  = 1  + (0.110 + 0.993i)2-s + (−0.975 + 0.220i)4-s + (−0.745 − 0.666i)5-s + (−0.0792 + 0.996i)7-s + (−0.327 − 0.945i)8-s + (0.580 − 0.814i)10-s + (−0.553 − 0.832i)11-s + (−0.823 + 0.567i)13-s + (−0.999 + 0.0317i)14-s + (0.902 − 0.429i)16-s + (0.0475 − 0.998i)17-s + (0.0475 + 0.998i)19-s + (0.873 + 0.486i)20-s + (0.766 − 0.642i)22-s + (0.110 + 0.993i)25-s + (−0.654 − 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $0.920 + 0.391i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ 0.920 + 0.391i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8496747924 + 0.1732110415i\)
\(L(\frac12)\) \(\approx\) \(0.8496747924 + 0.1732110415i\)
\(L(1)\) \(\approx\) \(0.7599318421 + 0.2910699053i\)
\(L(1)\) \(\approx\) \(0.7599318421 + 0.2910699053i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.110 + 0.993i)T \)
5 \( 1 + (-0.745 - 0.666i)T \)
7 \( 1 + (-0.0792 + 0.996i)T \)
11 \( 1 + (-0.553 - 0.832i)T \)
13 \( 1 + (-0.823 + 0.567i)T \)
17 \( 1 + (0.0475 - 0.998i)T \)
19 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (0.678 - 0.734i)T \)
31 \( 1 + (0.873 - 0.486i)T \)
37 \( 1 + (0.928 - 0.371i)T \)
41 \( 1 + (0.950 - 0.312i)T \)
43 \( 1 + (-0.0158 - 0.999i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (0.902 + 0.429i)T \)
61 \( 1 + (0.356 - 0.934i)T \)
67 \( 1 + (-0.444 - 0.895i)T \)
71 \( 1 + (0.723 + 0.690i)T \)
73 \( 1 + (0.0475 + 0.998i)T \)
79 \( 1 + (0.967 - 0.251i)T \)
83 \( 1 + (0.950 + 0.312i)T \)
89 \( 1 + (0.981 - 0.189i)T \)
97 \( 1 + (0.527 - 0.849i)T \)
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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

−22.95108402565508521379315038717, −22.117803507252165185352887956463, −21.30738896709433552638745040383, −20.21709452148586292977955341481, −19.724954654506012380161655873333, −19.19630532566728747581096276747, −17.84901041347251596156324215231, −17.65025427038950520112235549354, −16.31928657865084911102552540633, −15.070038735368605690116877819965, −14.64013525080820272741083882316, −13.48860066203027891601749128546, −12.74427972592949472225451628478, −11.93719187316328712901516421480, −10.89063984698370317283320900145, −10.39947993184864490710925743590, −9.644627392195133851796114295881, −8.232258016897424119049393478965, −7.532549412093044427211363922189, −6.49951541376970287088502676747, −4.913218473383866747476662777526, −4.30555329711036215904164258953, −3.22042568573875774599739266629, −2.44888341329387503443564673297, −0.922654482499628265876200172971, 0.57252151598013127996704137259, 2.53903720087564786316078714677, 3.76173843578382273525919252739, 4.81965209890613335258552689805, 5.48943119252268549084275639039, 6.46405974098993115871686297270, 7.68252765547348408395285293896, 8.229710715454178068082960600298, 9.11522751778865379893800370812, 9.87548610300927083323549259732, 11.52095206225405424641176139558, 12.16200266644879958725387411580, 12.99710707596499641877866707925, 13.98481408943739702638966856643, 14.83702367037824624319731066441, 15.76585424147389710048700195023, 16.18796386369656056842523875810, 16.96492179934882406744561740421, 18.06761620227732205262890353458, 18.89583535954449292466645444960, 19.394995378961691251747434528954, 20.85185435902665708880714383396, 21.45923799218006794953672282388, 22.477667542484511073795552047967, 23.13240095580003631080093876134

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