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Properties

Label 1-1441-1441.1007-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.859 + 0.511i$
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  + (−0.681 + 0.732i)2-s + (0.995 − 0.0965i)3-s + (−0.0724 − 0.997i)4-s + (0.399 + 0.916i)5-s + (−0.607 + 0.794i)6-s + (−0.568 + 0.822i)7-s + (0.779 + 0.626i)8-s + (0.981 − 0.192i)9-s + (−0.943 − 0.331i)10-s + (−0.168 − 0.985i)12-s + (−0.568 + 0.822i)13-s + (−0.215 − 0.976i)14-s + (0.485 + 0.873i)15-s + (−0.989 + 0.144i)16-s + (−0.168 + 0.985i)17-s + (−0.527 + 0.849i)18-s + ⋯
L(s)  = 1  + (−0.681 + 0.732i)2-s + (0.995 − 0.0965i)3-s + (−0.0724 − 0.997i)4-s + (0.399 + 0.916i)5-s + (−0.607 + 0.794i)6-s + (−0.568 + 0.822i)7-s + (0.779 + 0.626i)8-s + (0.981 − 0.192i)9-s + (−0.943 − 0.331i)10-s + (−0.168 − 0.985i)12-s + (−0.568 + 0.822i)13-s + (−0.215 − 0.976i)14-s + (0.485 + 0.873i)15-s + (−0.989 + 0.144i)16-s + (−0.168 + 0.985i)17-s + (−0.527 + 0.849i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.859 + 0.511i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.859 + 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3922459499 + 1.424507964i\)
\(L(\frac12)\) \(\approx\) \(0.3922459499 + 1.424507964i\)
\(L(1)\) \(\approx\) \(0.8504433134 + 0.6548859783i\)
\(L(1)\) \(\approx\) \(0.8504433134 + 0.6548859783i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.681 + 0.732i)T \)
3 \( 1 + (0.995 - 0.0965i)T \)
5 \( 1 + (0.399 + 0.916i)T \)
7 \( 1 + (-0.568 + 0.822i)T \)
13 \( 1 + (-0.568 + 0.822i)T \)
17 \( 1 + (-0.168 + 0.985i)T \)
19 \( 1 + (0.715 + 0.698i)T \)
23 \( 1 + (0.262 - 0.964i)T \)
29 \( 1 + (0.120 + 0.992i)T \)
31 \( 1 + (0.998 + 0.0483i)T \)
37 \( 1 + (0.970 + 0.239i)T \)
41 \( 1 + (0.443 - 0.896i)T \)
43 \( 1 + (-0.399 - 0.916i)T \)
47 \( 1 + (-0.981 + 0.192i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.168 + 0.985i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.399 + 0.916i)T \)
71 \( 1 + (-0.836 + 0.548i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.215 - 0.976i)T \)
83 \( 1 + (-0.0724 + 0.997i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.958 - 0.285i)T \)
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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

−20.298689122671477137866239495574, −19.69178492875794109711771949263, −19.37168177428347078032617949026, −18.09112006916621771636234149395, −17.605128965863422484385813863725, −16.674290761329090707532626934980, −16.05667643595117369138229988171, −15.301309312753385919205898478238, −13.93147580877540164375595469199, −13.38328891008119220542513853563, −12.977204639802110976809102598123, −12.02825045712973986924175719753, −11.07475812929978304581974982345, −9.84589178753951557245902363758, −9.734512443299180882591058792312, −9.05223893832036585485079262349, −7.85115244118996456341978234370, −7.67167018332450620649556083591, −6.493393737994215465082706759957, −4.9130703343224051830714103732, −4.32624084505983416788064034039, −3.14820790557628552636579927802, −2.69255270441382992236233010921, −1.44392752791203222087437595782, −0.6284451577719554294827933291, 1.51984464034861795292479891702, 2.30863956656691381756566616900, 3.107607727765626781721559530143, 4.29687811010171133476778583276, 5.4990982942008359644185725251, 6.48151256135943725054816612156, 6.85497051974201918136247445742, 7.84947156093219440892229409499, 8.62357235637676943179051711220, 9.356972014285140715238260596408, 9.96628338684617726385222147974, 10.62612999157873725326624596899, 11.8480022468063916859846591661, 12.84752132668271938201156227803, 13.76611030422070271781378916722, 14.56276101369350409819548101372, 14.80364054785893491441869806745, 15.7087646519792771532428780173, 16.377359524571671898353477845814, 17.37627556068951170640320889065, 18.25526524366322129719421564547, 18.80647663298279896139874898788, 19.20953850277383027221586810595, 19.971038584881285516076979211714, 20.99006899349104263697499412999

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