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Properties

Label 1-1441-1441.692-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.407 - 0.913i$
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  + (0.399 − 0.916i)2-s + (−0.262 − 0.964i)3-s + (−0.681 − 0.732i)4-s + (−0.998 − 0.0483i)5-s + (−0.989 − 0.144i)6-s + (0.168 − 0.985i)7-s + (−0.943 + 0.331i)8-s + (−0.861 + 0.506i)9-s + (−0.443 + 0.896i)10-s + (−0.527 + 0.849i)12-s + (0.443 + 0.896i)13-s + (−0.836 − 0.548i)14-s + (0.215 + 0.976i)15-s + (−0.0724 + 0.997i)16-s + (0.926 + 0.377i)17-s + (0.120 + 0.992i)18-s + ⋯
L(s)  = 1  + (0.399 − 0.916i)2-s + (−0.262 − 0.964i)3-s + (−0.681 − 0.732i)4-s + (−0.998 − 0.0483i)5-s + (−0.989 − 0.144i)6-s + (0.168 − 0.985i)7-s + (−0.943 + 0.331i)8-s + (−0.861 + 0.506i)9-s + (−0.443 + 0.896i)10-s + (−0.527 + 0.849i)12-s + (0.443 + 0.896i)13-s + (−0.836 − 0.548i)14-s + (0.215 + 0.976i)15-s + (−0.0724 + 0.997i)16-s + (0.926 + 0.377i)17-s + (0.120 + 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.407 - 0.913i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (692, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.407 - 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7649560257 - 0.4964353991i\)
\(L(\frac12)\) \(\approx\) \(0.7649560257 - 0.4964353991i\)
\(L(1)\) \(\approx\) \(0.6273473769 - 0.5866087600i\)
\(L(1)\) \(\approx\) \(0.6273473769 - 0.5866087600i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.399 - 0.916i)T \)
3 \( 1 + (-0.262 - 0.964i)T \)
5 \( 1 + (-0.998 - 0.0483i)T \)
7 \( 1 + (0.168 - 0.985i)T \)
13 \( 1 + (0.443 + 0.896i)T \)
17 \( 1 + (0.926 + 0.377i)T \)
19 \( 1 + (-0.970 - 0.239i)T \)
23 \( 1 + (-0.0241 + 0.999i)T \)
29 \( 1 + (-0.861 - 0.506i)T \)
31 \( 1 + (0.607 + 0.794i)T \)
37 \( 1 + (0.906 - 0.421i)T \)
41 \( 1 + (0.0724 + 0.997i)T \)
43 \( 1 + (-0.779 - 0.626i)T \)
47 \( 1 + (0.748 + 0.663i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.644 - 0.764i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.779 + 0.626i)T \)
71 \( 1 + (-0.568 + 0.822i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.354 + 0.935i)T \)
83 \( 1 + (0.981 - 0.192i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.715 + 0.698i)T \)
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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

−20.835394583890234801522851372946, −20.56193863413385659164574191386, −19.18066703157455665624451695640, −18.47145104062330444577713302291, −17.73999123231457305070906224134, −16.61700982274127152500135067240, −16.353499838355815809153629187873, −15.446838714428396796663630656221, −14.90197866962520075027377768117, −14.64600014004365173623082528518, −13.27270721243333249446987264177, −12.347163322687876754844938659627, −11.85807509174117392831982562824, −10.93924140359822913033406516657, −9.99204642047301036068660466037, −8.94124418823543455415760163392, −8.38101490128963028672091319270, −7.737968289717856343508136625839, −6.5074031102855367523870410938, −5.7435736703374802172371429586, −5.057724954611417233448733444017, −4.236972964062394931220703395543, −3.46132442255183480068039715267, −2.68788673247297966756818606240, −0.41238244849772504262671133330, 0.9354947987412922128665253343, 1.58449617718021863687221952526, 2.79366302624431050432617224881, 3.8206038911863014603959678340, 4.35322753214882393794269493160, 5.45790939406262804741465543625, 6.429124773494272318752442049851, 7.28357795035499252239268904475, 8.08512824877626427661387753005, 8.86752301842181232941310632957, 10.05559165092221981728597726371, 10.971342698948312544271634329547, 11.43824948031642540034140996710, 12.08512679999349614055731328582, 12.93933032591279977909826342927, 13.48913190243784352348500662516, 14.323274018048745939439002173789, 14.9456992002859805425377023240, 16.17459649982789683782537129229, 16.998640931990992490160498182010, 17.69705361525610892138128606013, 18.77844303931823323210311217517, 19.103434763588138442645231503185, 19.75581409604777957917097122731, 20.42679740820585256136533903038

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