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Properties

Label 1-1441-1441.684-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.227 + 0.973i$
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  + (−0.998 + 0.0483i)2-s + (0.958 + 0.285i)3-s + (0.995 − 0.0965i)4-s + (−0.943 + 0.331i)5-s + (−0.970 − 0.239i)6-s + (−0.644 + 0.764i)7-s + (−0.989 + 0.144i)8-s + (0.836 + 0.548i)9-s + (0.926 − 0.377i)10-s + (0.981 + 0.192i)12-s + (0.527 + 0.849i)13-s + (0.607 − 0.794i)14-s + (−0.998 + 0.0483i)15-s + (0.981 − 0.192i)16-s + (0.120 − 0.992i)17-s + (−0.861 − 0.506i)18-s + ⋯
L(s)  = 1  + (−0.998 + 0.0483i)2-s + (0.958 + 0.285i)3-s + (0.995 − 0.0965i)4-s + (−0.943 + 0.331i)5-s + (−0.970 − 0.239i)6-s + (−0.644 + 0.764i)7-s + (−0.989 + 0.144i)8-s + (0.836 + 0.548i)9-s + (0.926 − 0.377i)10-s + (0.981 + 0.192i)12-s + (0.527 + 0.849i)13-s + (0.607 − 0.794i)14-s + (−0.998 + 0.0483i)15-s + (0.981 − 0.192i)16-s + (0.120 − 0.992i)17-s + (−0.861 − 0.506i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.227 + 0.973i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (684, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.227 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9488199849 + 0.7529411298i\)
\(L(\frac12)\) \(\approx\) \(0.9488199849 + 0.7529411298i\)
\(L(1)\) \(\approx\) \(0.8320637447 + 0.2817275031i\)
\(L(1)\) \(\approx\) \(0.8320637447 + 0.2817275031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.998 + 0.0483i)T \)
3 \( 1 + (0.958 + 0.285i)T \)
5 \( 1 + (-0.943 + 0.331i)T \)
7 \( 1 + (-0.644 + 0.764i)T \)
13 \( 1 + (0.527 + 0.849i)T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (0.981 + 0.192i)T \)
23 \( 1 + (0.989 + 0.144i)T \)
29 \( 1 + (-0.262 - 0.964i)T \)
31 \( 1 + (0.989 - 0.144i)T \)
37 \( 1 + (-0.215 + 0.976i)T \)
41 \( 1 + (-0.120 - 0.992i)T \)
43 \( 1 + (0.607 + 0.794i)T \)
47 \( 1 + (0.998 - 0.0483i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.681 + 0.732i)T \)
61 \( 1 - T \)
67 \( 1 + (0.607 - 0.794i)T \)
71 \( 1 + (0.989 - 0.144i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.943 + 0.331i)T \)
83 \( 1 + (0.215 + 0.976i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.644 + 0.764i)T \)
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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

−20.23356149562357604424830410145, −19.84791780192032482994224309562, −19.1152996801011682749924463156, −18.60236768290565536299174407762, −17.60874528362485663150558616425, −16.82796781719091435958867074532, −15.92843210632661005111326094776, −15.519589784199342651382383212060, −14.72926640120200053799071748513, −13.66258126811050813086007928754, −12.72097250819550638399773238481, −12.339772047013631598152077533516, −11.06756626030018910488199136206, −10.485868449272481418643696847012, −9.566585313125519851267698622148, −8.78495788647065200461628128450, −8.18090145666495910973589676576, −7.38240259072451835304472345540, −6.966282137316867607231901249770, −5.79376327745172367751051181851, −4.27311027607973202306053225141, −3.343585780897729914463953297538, −2.94521240186723926363049970518, −1.39710051444837262354365987999, −0.74367170340856204832780206005, 1.02205345272067325497292775159, 2.359346893228910098898576095739, 3.00182518818490800437735426290, 3.72965173868077921247147707943, 4.960344629111377653890932818994, 6.26632814502781052448736917347, 7.09630626387908769420290115370, 7.72685478170526390778644928147, 8.57485749730568712272392156845, 9.22558979883520663830166414106, 9.76737993249532144626615745130, 10.75918301639454678371399580962, 11.67027320887341916093435973517, 12.13494321137970813367458764582, 13.38533905736935358262372644630, 14.2465198901143061693579366206, 15.20155009989709138113908006169, 15.67437395767685208331275955316, 16.11996291199615404264292152673, 16.98380736869571367059952552011, 18.42117992550857837504512326684, 18.65701132547542059893408666082, 19.27932166585709362989199434913, 19.90232611929797750618818238419, 20.76893072486438318784872016108

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