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Properties

Label 1-1441-1441.180-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.402 - 0.915i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (−0.443 − 0.896i)2-s + (0.926 − 0.377i)3-s + (−0.607 + 0.794i)4-s + (−0.0724 − 0.997i)5-s + (−0.748 − 0.663i)6-s + (0.399 + 0.916i)7-s + (0.981 + 0.192i)8-s + (0.715 − 0.698i)9-s + (−0.861 + 0.506i)10-s + (−0.262 + 0.964i)12-s + (0.215 − 0.976i)13-s + (0.644 − 0.764i)14-s + (−0.443 − 0.896i)15-s + (−0.262 − 0.964i)16-s + (−0.354 + 0.935i)17-s + (−0.943 − 0.331i)18-s + ⋯
L(s)  = 1  + (−0.443 − 0.896i)2-s + (0.926 − 0.377i)3-s + (−0.607 + 0.794i)4-s + (−0.0724 − 0.997i)5-s + (−0.748 − 0.663i)6-s + (0.399 + 0.916i)7-s + (0.981 + 0.192i)8-s + (0.715 − 0.698i)9-s + (−0.861 + 0.506i)10-s + (−0.262 + 0.964i)12-s + (0.215 − 0.976i)13-s + (0.644 − 0.764i)14-s + (−0.443 − 0.896i)15-s + (−0.262 − 0.964i)16-s + (−0.354 + 0.935i)17-s + (−0.943 − 0.331i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.402 - 0.915i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.402 - 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9688557902 - 1.483494390i\)
\(L(\frac12)\) \(\approx\) \(0.9688557902 - 1.483494390i\)
\(L(1)\) \(\approx\) \(0.9736842548 - 0.7069245591i\)
\(L(1)\) \(\approx\) \(0.9736842548 - 0.7069245591i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.443 - 0.896i)T \)
3 \( 1 + (0.926 - 0.377i)T \)
5 \( 1 + (-0.0724 - 0.997i)T \)
7 \( 1 + (0.399 + 0.916i)T \)
13 \( 1 + (0.215 - 0.976i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (-0.262 + 0.964i)T \)
23 \( 1 + (0.981 - 0.192i)T \)
29 \( 1 + (-0.168 - 0.985i)T \)
31 \( 1 + (0.981 + 0.192i)T \)
37 \( 1 + (0.958 - 0.285i)T \)
41 \( 1 + (-0.354 - 0.935i)T \)
43 \( 1 + (0.644 + 0.764i)T \)
47 \( 1 + (-0.443 - 0.896i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.998 - 0.0483i)T \)
61 \( 1 + T \)
67 \( 1 + (0.644 - 0.764i)T \)
71 \( 1 + (0.981 + 0.192i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.0724 - 0.997i)T \)
83 \( 1 + (0.958 + 0.285i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.399 + 0.916i)T \)
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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

−20.88131647696245102027890957964, −20.00270455229303913361904996984, −19.352788949626046266531444610225, −18.70182505717714682341023137589, −17.99874534502896539820850512071, −17.13319657890837051311517207078, −16.35198077407794773415087837219, −15.55191740736802344064863617926, −14.96986662164432679233103595894, −14.14447303217564302384596296311, −13.854043860553143578972123734337, −13.077221298554698380286762664716, −11.21464853957241036684348638567, −10.94160443376395064003884403619, −9.878864929542296553420907810205, −9.32967470502442693210162250548, −8.47164381905091012879514929727, −7.56235662963545017502351564621, −7.04185559457146364844379291804, −6.43224406393893654025853802622, −4.87185812934435755126739209002, −4.41243251480818774187824320228, −3.35121331866603387767293213280, −2.32386165861441457414699361821, −1.1392689263603072601933629850, 0.82030053489497629328064652356, 1.74023792635377898550184660874, 2.4337810019127704781837325596, 3.445626275466288275957581756887, 4.26526814519519138514939561794, 5.21732715463055096409675856898, 6.31446960043639531319130423179, 7.90078281171126165590940752830, 8.10956999513747060098816602906, 8.80711786471595536735640778326, 9.49130074171318434551188193642, 10.3256323333641830668504637577, 11.38838065970899288079542338811, 12.259935526082731204389672729561, 12.7906069449763371914867961174, 13.24607227191033149200357534532, 14.30127049567655264368809814967, 15.187831917484974586962178163942, 15.87144230446216953303861674150, 17.06563216539622747715282199075, 17.60257760670468931840955019818, 18.489257108975317439108196962870, 19.09119879291996705448648878039, 19.75240790756316626598621800238, 20.46258852177424674114784446770

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