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Properties

Label 1-920-920.867-r0-0-0
Degree $1$
Conductor $920$
Sign $-0.736 + 0.676i$
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  + (−0.909 − 0.415i)3-s + (−0.540 − 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.540 + 0.841i)13-s + (0.281 − 0.959i)17-s + (0.959 − 0.281i)19-s + (0.142 + 0.989i)21-s + (−0.281 − 0.959i)27-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.540 − 0.841i)33-s + (−0.755 + 0.654i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)3-s + (−0.540 − 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (−0.540 + 0.841i)13-s + (0.281 − 0.959i)17-s + (0.959 − 0.281i)19-s + (0.142 + 0.989i)21-s + (−0.281 − 0.959i)27-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.540 − 0.841i)33-s + (−0.755 + 0.654i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.736 + 0.676i$
Analytic conductor: \(4.27246\)
Root analytic conductor: \(4.27246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 920,\ (0:\ ),\ -0.736 + 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04452910912 + 0.1142300228i\)
\(L(\frac12)\) \(\approx\) \(0.04452910912 + 0.1142300228i\)
\(L(1)\) \(\approx\) \(0.6019917190 - 0.06823475360i\)
\(L(1)\) \(\approx\) \(0.6019917190 - 0.06823475360i\)

Euler product

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

−21.689729116831627046498784059218, −21.090505523982021811437262327919, −19.982388824234538444608398128116, −19.091192305087564611292446716870, −18.35043167439756214376948215570, −17.63705001701650401826811464957, −16.67902145652240798559713421129, −16.12382160935812402935229017573, −15.34872685834713378866360207826, −14.62710054204990059189761469350, −13.37805698640335309655529816917, −12.49919370107618060385738889936, −12.00438034555669881880417011901, −10.969064348293825854778480831045, −10.31075704642740942160000267439, −9.44383120124238983437136948861, −8.5859309555209208291297329749, −7.486073708276884180300195093129, −6.3932764267565552220297486081, −5.5764211102378642889462998907, −5.201565930091406196715886016175, −3.688761776593216121673367625342, −3.07196985140207633379890794253, −1.53392412329630936708800419971, −0.064360194894936529323468940469, 1.2809498735162011836558209013, 2.36340577399336553894647852044, 3.742806290574900057902073456437, 4.74134021315041077424364672128, 5.41948656504410138479838442000, 6.77598359084325535467175954497, 7.05746355626782646871184490910, 7.889662582503311964424128558005, 9.61498383692364217490684941702, 9.805363675811993951247629682479, 10.98708869982189396320124203676, 11.75260132743316075653942357795, 12.41985540211107168381163190655, 13.3850205728026253285393955135, 13.907564129941860022759298439638, 15.13788094418939346891366138203, 16.02513168169897295428125206977, 16.875032824482974917644295054, 17.18381568620532075514869654479, 18.38614756658272463513510287579, 18.725938685158346936600315489372, 19.96267307144834952915056711798, 20.39004638486861381012849520968, 21.59575989866933385858364806149, 22.39961246162652673277834417093

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