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Properties

Label 1-1441-1441.934-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.193 + 0.981i$
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  + (0.485 − 0.873i)2-s + (0.399 − 0.916i)3-s + (−0.527 − 0.849i)4-s + (0.215 + 0.976i)5-s + (−0.607 − 0.794i)6-s + (−0.0241 + 0.999i)7-s + (−0.998 + 0.0483i)8-s + (−0.681 − 0.732i)9-s + (0.958 + 0.285i)10-s + (−0.989 + 0.144i)12-s + (−0.958 + 0.285i)13-s + (0.861 + 0.506i)14-s + (0.981 + 0.192i)15-s + (−0.443 + 0.896i)16-s + (−0.168 − 0.985i)17-s + (−0.970 + 0.239i)18-s + ⋯
L(s)  = 1  + (0.485 − 0.873i)2-s + (0.399 − 0.916i)3-s + (−0.527 − 0.849i)4-s + (0.215 + 0.976i)5-s + (−0.607 − 0.794i)6-s + (−0.0241 + 0.999i)7-s + (−0.998 + 0.0483i)8-s + (−0.681 − 0.732i)9-s + (0.958 + 0.285i)10-s + (−0.989 + 0.144i)12-s + (−0.958 + 0.285i)13-s + (0.861 + 0.506i)14-s + (0.981 + 0.192i)15-s + (−0.443 + 0.896i)16-s + (−0.168 − 0.985i)17-s + (−0.970 + 0.239i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.193 + 0.981i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (934, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.193 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2177594085 + 0.1789563012i\)
\(L(\frac12)\) \(\approx\) \(0.2177594085 + 0.1789563012i\)
\(L(1)\) \(\approx\) \(0.8963477717 - 0.5077920371i\)
\(L(1)\) \(\approx\) \(0.8963477717 - 0.5077920371i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.485 - 0.873i)T \)
3 \( 1 + (0.399 - 0.916i)T \)
5 \( 1 + (0.215 + 0.976i)T \)
7 \( 1 + (-0.0241 + 0.999i)T \)
13 \( 1 + (-0.958 + 0.285i)T \)
17 \( 1 + (-0.168 - 0.985i)T \)
19 \( 1 + (0.885 + 0.464i)T \)
23 \( 1 + (-0.779 - 0.626i)T \)
29 \( 1 + (-0.681 + 0.732i)T \)
31 \( 1 + (-0.836 - 0.548i)T \)
37 \( 1 + (-0.926 - 0.377i)T \)
41 \( 1 + (0.443 + 0.896i)T \)
43 \( 1 + (-0.995 - 0.0965i)T \)
47 \( 1 + (-0.120 - 0.992i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.715 + 0.698i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.995 + 0.0965i)T \)
71 \( 1 + (0.354 + 0.935i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (0.644 + 0.764i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.943 + 0.331i)T \)
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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

−20.64380543241216573303352506547, −20.03763682481707138560091273002, −19.3668262434548300406075731297, −17.79743246380837025118601418136, −17.182355897616727172173456678836, −16.78479264123329672745986702962, −15.91513801694753863123015860337, −15.43647346224066516871189826045, −14.44639587788903926486022589075, −13.87918897675929520474775952121, −13.1950625011838413704486186255, −12.414623885957241899961173408482, −11.41899204645248240651734004105, −10.27159123905044106552933309966, −9.564882228382404593667574586941, −8.911048964994638056225086813736, −7.90909443799711721879147576017, −7.50616563497800104489604348208, −6.19656591181993457326377640030, −5.26870125982668287687562395864, −4.745296470562532712041050020740, −3.92429862603982298483929113800, −3.26708132873074610983150449314, −1.84620077589532959601601271589, −0.076084364945814357323517631485, 1.62321942304280990490105846774, 2.329060376874429126969982552471, 2.89223713800850512360884547323, 3.72815971961719723175187623488, 5.16175047587592395138756646157, 5.80400155318010993133142083452, 6.729831116665023131591122151462, 7.46424841150197126298530050193, 8.61358289060360785632064659563, 9.46289793610947387543515620008, 10.01768438444343088396477875749, 11.26710159158105995875770895634, 11.78591617866529770538200473794, 12.389017318361122992674649696745, 13.24719054377870330760450984144, 14.046438966347675994549115523812, 14.60566166856309017368858551486, 15.07665998872801356345212271874, 16.273078642615542231884498345127, 17.66900156980742992378425582049, 18.41163415518755205882596312749, 18.47150674498176899146612266023, 19.39848335652263438170483808961, 20.04880835765375849495659660739, 20.82543052275472723277358914955

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