Properties

Label 1-1441-1441.1020-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.907 - 0.420i$
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  + (−0.943 − 0.331i)2-s + (−0.443 − 0.896i)3-s + (0.779 + 0.626i)4-s + (0.715 − 0.698i)5-s + (0.120 + 0.992i)6-s + (−0.981 + 0.192i)7-s + (−0.527 − 0.849i)8-s + (−0.607 + 0.794i)9-s + (−0.906 + 0.421i)10-s + (0.215 − 0.976i)12-s + (0.681 − 0.732i)13-s + (0.989 + 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.215 + 0.976i)16-s + (−0.748 − 0.663i)17-s + (0.836 − 0.548i)18-s + ⋯
L(s)  = 1  + (−0.943 − 0.331i)2-s + (−0.443 − 0.896i)3-s + (0.779 + 0.626i)4-s + (0.715 − 0.698i)5-s + (0.120 + 0.992i)6-s + (−0.981 + 0.192i)7-s + (−0.527 − 0.849i)8-s + (−0.607 + 0.794i)9-s + (−0.906 + 0.421i)10-s + (0.215 − 0.976i)12-s + (0.681 − 0.732i)13-s + (0.989 + 0.144i)14-s + (−0.943 − 0.331i)15-s + (0.215 + 0.976i)16-s + (−0.748 − 0.663i)17-s + (0.836 − 0.548i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.907 - 0.420i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1020, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.907 - 0.420i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1746040954 - 0.7914173231i\)
\(L(\frac12)\) \(\approx\) \(0.1746040954 - 0.7914173231i\)
\(L(1)\) \(\approx\) \(0.5220533311 - 0.4027896374i\)
\(L(1)\) \(\approx\) \(0.5220533311 - 0.4027896374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.943 - 0.331i)T \)
3 \( 1 + (-0.443 - 0.896i)T \)
5 \( 1 + (0.715 - 0.698i)T \)
7 \( 1 + (-0.981 + 0.192i)T \)
13 \( 1 + (0.681 - 0.732i)T \)
17 \( 1 + (-0.748 - 0.663i)T \)
19 \( 1 + (0.215 - 0.976i)T \)
23 \( 1 + (0.527 - 0.849i)T \)
29 \( 1 + (0.958 + 0.285i)T \)
31 \( 1 + (0.527 + 0.849i)T \)
37 \( 1 + (0.998 + 0.0483i)T \)
41 \( 1 + (0.748 - 0.663i)T \)
43 \( 1 + (0.989 - 0.144i)T \)
47 \( 1 + (0.943 + 0.331i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.861 + 0.506i)T \)
61 \( 1 - T \)
67 \( 1 + (0.989 + 0.144i)T \)
71 \( 1 + (0.527 + 0.849i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.715 - 0.698i)T \)
83 \( 1 + (-0.998 + 0.0483i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.981 + 0.192i)T \)
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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

−21.18083665792855589922638307590, −20.21936657398955779633890042251, −19.40859893386344824889855501631, −18.68176523969580619030148385588, −17.957306688588614267144263396847, −17.11792174065497596633810617496, −16.72785802741017619848975643526, −15.76958284192230949351877407514, −15.38960081223214097521219931013, −14.391189428539624257075079301252, −13.71009084315035995220781931469, −12.5250664034036809169567243840, −11.333779455960282021437470654295, −10.92641760621948570521975032118, −10.04973783370806780443612035959, −9.57881619679872927562097851179, −8.97686345669138505312509138105, −7.839388321019183137804144840191, −6.65777985617834517009552208767, −6.23292579367427620004142974448, −5.66684667385937235946472228114, −4.28440519986744806499350802493, −3.304920199893450117810881618067, −2.375280226935384204970993106374, −1.110361429654751289848266526841, 0.565575720485527929119299322814, 1.16444846261034522052517157326, 2.53395135788334903033289538441, 2.84285797362973015431842790456, 4.53194143147853882211244790019, 5.68453749850885252089826560034, 6.41301262560097961946481155818, 6.989121447877930167448623208215, 8.047087048551942276366221039550, 8.90659514040885671752749146965, 9.32706228745761024690881622098, 10.505545745132148347172978889370, 10.979974468872860102738486597733, 12.16023706613583194399869173233, 12.60813770288138271472932017402, 13.26392095925955084780375060402, 13.95232996435426178564061359025, 15.641931115185218638914541435705, 16.01714466029814639604494890902, 16.90375782506237756197797698787, 17.54629812809779795276473110303, 18.11462946798060242516224995181, 18.740911691758795132001584986069, 19.71656421124087868688359156469, 20.06149694715515506903878715354

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