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Properties

Label 1-1441-1441.1028-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.516 + 0.856i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (−0.120 + 0.992i)2-s + (0.215 − 0.976i)3-s + (−0.970 − 0.239i)4-s + (−0.861 − 0.506i)5-s + (0.943 + 0.331i)6-s + (0.0241 + 0.999i)7-s + (0.354 − 0.935i)8-s + (−0.906 − 0.421i)9-s + (0.607 − 0.794i)10-s + (−0.443 + 0.896i)12-s + (0.958 + 0.285i)13-s + (−0.995 − 0.0965i)14-s + (−0.681 + 0.732i)15-s + (0.885 + 0.464i)16-s + (0.989 − 0.144i)17-s + (0.527 − 0.849i)18-s + ⋯
L(s)  = 1  + (−0.120 + 0.992i)2-s + (0.215 − 0.976i)3-s + (−0.970 − 0.239i)4-s + (−0.861 − 0.506i)5-s + (0.943 + 0.331i)6-s + (0.0241 + 0.999i)7-s + (0.354 − 0.935i)8-s + (−0.906 − 0.421i)9-s + (0.607 − 0.794i)10-s + (−0.443 + 0.896i)12-s + (0.958 + 0.285i)13-s + (−0.995 − 0.0965i)14-s + (−0.681 + 0.732i)15-s + (0.885 + 0.464i)16-s + (0.989 − 0.144i)17-s + (0.527 − 0.849i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1028, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.516 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103059730 + 0.6231724483i\)
\(L(\frac12)\) \(\approx\) \(1.103059730 + 0.6231724483i\)
\(L(1)\) \(\approx\) \(0.8263523765 + 0.1433974903i\)
\(L(1)\) \(\approx\) \(0.8263523765 + 0.1433974903i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.120 + 0.992i)T \)
3 \( 1 + (0.215 - 0.976i)T \)
5 \( 1 + (-0.861 - 0.506i)T \)
7 \( 1 + (0.0241 + 0.999i)T \)
13 \( 1 + (0.958 + 0.285i)T \)
17 \( 1 + (0.989 - 0.144i)T \)
19 \( 1 + (-0.715 - 0.698i)T \)
23 \( 1 + (0.998 - 0.0483i)T \)
29 \( 1 + (0.681 + 0.732i)T \)
31 \( 1 + (-0.779 - 0.626i)T \)
37 \( 1 + (-0.926 + 0.377i)T \)
41 \( 1 + (-0.168 - 0.985i)T \)
43 \( 1 + (0.215 - 0.976i)T \)
47 \( 1 + (-0.981 + 0.192i)T \)
53 \( 1 + T \)
59 \( 1 + (0.885 + 0.464i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.215 - 0.976i)T \)
71 \( 1 + (-0.836 + 0.548i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.215 + 0.976i)T \)
83 \( 1 + (0.527 + 0.849i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.607 + 0.794i)T \)
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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

−20.60366998651693151707789765838, −19.57052638799911856246084734703, −19.331812379346881329115017199512, −18.341703977471198928719229700426, −17.43542406903564522206856005182, −16.59152906097079286314948980189, −16.0341852480335478908347740214, −14.82945184593001530703250083989, −14.46306769386156994134695900495, −13.55018028181674118323585290985, −12.709367711038500981857540083396, −11.64543535100009537705272461841, −11.068590135187517384581267848703, −10.38417079586696121605897315280, −10.01314314735234685573889048222, −8.772876319109538180797925413206, −8.20346616136879637782890054055, −7.403502078459661164361598630307, −6.057691558633083122063855785589, −4.87243572504935531564816908868, −4.1482995165971429281039051230, −3.446117130294522405884018467614, −3.03307431418463940545266870949, −1.5160462269461638060397413776, −0.39821868330675467846861094931, 0.669036101279625702590226959479, 1.57379295576754010918039480219, 2.99558433376308696707277634196, 3.90485948839838818823952969082, 5.1141745829013879816283190021, 5.69724964881012149107281698327, 6.72157042025346038501473218749, 7.2896545202087822874452790585, 8.28990373064385324735672398536, 8.704710906005263149604560894173, 9.225848752637301445916532993521, 10.733424227050637149796632975, 11.74152659560316153963667452955, 12.42931642608743282800534418788, 13.06610197554494835431528114480, 13.84009274572926742235723012303, 14.82973047614275368789907735522, 15.26423588311210838159429180316, 16.160144277366501896168488941355, 16.81220000737050327878575207470, 17.694898815754481037914803060621, 18.48311843396587125407204388593, 19.0080458044019754023096667983, 19.4698951139468656255424252308, 20.61329161000587034646168815699

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