Properties

Label 1-920-920.629-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.820 - 0.571i$
Analytic cond. $4.27246$
Root an. cond. $4.27246$
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  + (−0.654 + 0.755i)3-s + (−0.415 − 0.909i)7-s + (−0.142 − 0.989i)9-s + (0.959 − 0.281i)11-s + (0.415 − 0.909i)13-s + (−0.841 + 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (0.841 + 0.540i)27-s + (−0.841 + 0.540i)29-s + (−0.654 − 0.755i)31-s + (−0.415 + 0.909i)33-s + (−0.142 − 0.989i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)3-s + (−0.415 − 0.909i)7-s + (−0.142 − 0.989i)9-s + (0.959 − 0.281i)11-s + (0.415 − 0.909i)13-s + (−0.841 + 0.540i)17-s + (−0.841 − 0.540i)19-s + (0.959 + 0.281i)21-s + (0.841 + 0.540i)27-s + (−0.841 + 0.540i)29-s + (−0.654 − 0.755i)31-s + (−0.415 + 0.909i)33-s + (−0.142 − 0.989i)37-s + (0.415 + 0.909i)39-s + (−0.142 + 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.820 - 0.571i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.820 - 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09018151787 - 0.2874670975i\)
\(L(\frac12)\) \(\approx\) \(0.09018151787 - 0.2874670975i\)
\(L(1)\) \(\approx\) \(0.6751604392 + 0.01893405926i\)
\(L(1)\) \(\approx\) \(0.6751604392 + 0.01893405926i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (-0.841 + 0.540i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (-0.142 - 0.989i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 - T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (0.654 + 0.755i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−22.2696533678468911589460921378, −21.68910810170564781229113007425, −20.60302218510068414292979046199, −19.56102998531282874734526717393, −18.948136568020005536192166926935, −18.36347686264780995829098428233, −17.47459788406010922354274015171, −16.72599094016711731636672802742, −16.01674810924939248931172956452, −15.03279247848573768148440942156, −14.11098996913026170103698835545, −13.23709583077416970787028113393, −12.47679590994900703764127984026, −11.719233876743943292409652906892, −11.2238681923113083409100405086, −10.00272117184241838785358247664, −9.00815811364228296062487411054, −8.37757948172137521715926848475, −6.98217699084335910950960724233, −6.57913561717384912652583637819, −5.72700603021006440664134428385, −4.73162710170232080437282391262, −3.634409853853315011759202369104, −2.20946207292198719160364038810, −1.609519798586369705286452451305, 0.14439006908118613817666338754, 1.42045057840368582546168168690, 3.12151764008926805468887519914, 3.94210126051846248884123292492, 4.57819250071300764948852635426, 5.86160934453339234663289596410, 6.42102189389002104656491487663, 7.388365300246073287417730583686, 8.68352370875705815771452221538, 9.36352834876309051904173707952, 10.39024959828127309687693763923, 10.901804916979862669185537489, 11.63052046684951362370732350880, 12.84757863942086550139064505118, 13.344045351981020464384709583352, 14.67196840605677966948272301931, 15.12609422647148731098508038624, 16.239832960629852424162089635236, 16.71422392537600273992597720802, 17.48361652194308080642917912204, 18.12915446360211351089090004274, 19.48152668565216914911872665771, 19.98582655923854265542188419640, 20.78765824439465650944137138972, 21.75091397750860264368638069453

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