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Properties

Label 1-1441-1441.31-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.966 - 0.257i$
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  + (0.443 − 0.896i)2-s + (0.644 − 0.764i)3-s + (−0.607 − 0.794i)4-s + (−0.527 − 0.849i)5-s + (−0.399 − 0.916i)6-s + (−0.748 − 0.663i)7-s + (−0.981 + 0.192i)8-s + (−0.168 − 0.985i)9-s + (−0.995 + 0.0965i)10-s + (−0.998 − 0.0483i)12-s + (−0.748 − 0.663i)13-s + (−0.926 + 0.377i)14-s + (−0.989 − 0.144i)15-s + (−0.262 + 0.964i)16-s + (0.998 − 0.0483i)17-s + (−0.958 − 0.285i)18-s + ⋯
L(s)  = 1  + (0.443 − 0.896i)2-s + (0.644 − 0.764i)3-s + (−0.607 − 0.794i)4-s + (−0.527 − 0.849i)5-s + (−0.399 − 0.916i)6-s + (−0.748 − 0.663i)7-s + (−0.981 + 0.192i)8-s + (−0.168 − 0.985i)9-s + (−0.995 + 0.0965i)10-s + (−0.998 − 0.0483i)12-s + (−0.748 − 0.663i)13-s + (−0.926 + 0.377i)14-s + (−0.989 − 0.144i)15-s + (−0.262 + 0.964i)16-s + (0.998 − 0.0483i)17-s + (−0.958 − 0.285i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.966 - 0.257i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.966 - 0.257i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3912893370 + 0.05116534936i\)
\(L(\frac12)\) \(\approx\) \(-0.3912893370 + 0.05116534936i\)
\(L(1)\) \(\approx\) \(0.4123068303 - 0.9329637676i\)
\(L(1)\) \(\approx\) \(0.4123068303 - 0.9329637676i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.443 - 0.896i)T \)
3 \( 1 + (0.644 - 0.764i)T \)
5 \( 1 + (-0.527 - 0.849i)T \)
7 \( 1 + (-0.748 - 0.663i)T \)
13 \( 1 + (-0.748 - 0.663i)T \)
17 \( 1 + (0.998 - 0.0483i)T \)
19 \( 1 + (-0.779 - 0.626i)T \)
23 \( 1 + (0.681 + 0.732i)T \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (0.906 + 0.421i)T \)
37 \( 1 + (-0.568 + 0.822i)T \)
41 \( 1 + (0.836 - 0.548i)T \)
43 \( 1 + (-0.527 - 0.849i)T \)
47 \( 1 + (0.168 + 0.985i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.998 + 0.0483i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.527 - 0.849i)T \)
71 \( 1 + (-0.485 - 0.873i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.926 - 0.377i)T \)
83 \( 1 + (0.607 - 0.794i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.861 - 0.506i)T \)
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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

−21.48515262662480396775536297622, −20.90163682781177207209913983205, −19.65962126450588309148896066376, −18.93419166938951843337820515837, −18.58961158353617635161019480906, −17.239106741904962419289884554916, −16.42616689220861548761168280526, −16.03585799597279394784979513142, −14.96951827252871372721926710499, −14.79553025169170507913629565435, −14.1499783695312631492272359513, −13.070841927273768076782187415696, −12.321738510975109721607502523730, −11.46580640206620032883582152176, −10.30985199526956329681268075494, −9.61874694558994022969090737666, −8.81249809346213772002468336952, −8.02548423053356738217176811930, −7.24638604257299794484322449528, −6.42437785398233276793326488794, −5.55242439694553895192466329166, −4.55608081981695061804592888033, −3.77337859485300966595155595527, −3.054663515466293971636556021424, −2.33207083478763223595555813199, 0.07450486209071294904822795529, 0.776532716954659414060644695872, 1.640240635043977088560996638881, 2.87502902458480878677023813290, 3.42171300304383099004071036909, 4.339213940526866227729668700775, 5.26226701477383877741505303274, 6.27207991611087777887227884779, 7.332258330792893897193839914680, 8.03235726522390452982466373377, 9.05988647292585749616305274326, 9.59848481688974941574285584879, 10.509797801739932885725933819140, 11.55813803074826886838116304104, 12.40044498627916014746952284355, 12.75242651235508557474240957845, 13.44735057399124282335737509783, 14.11455564192818995558126862495, 15.12363398978823852782065719017, 15.65489213445194142142959447424, 17.11519825348070758928747631026, 17.38078118255810364616187673630, 18.81001283524856420798111015404, 19.21528386474193694507319895092, 19.73020361121094135351400419030

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