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Properties

Label 1-1441-1441.335-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.775 - 0.631i$
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  + (−0.309 + 0.951i)2-s + 3-s + (−0.809 − 0.587i)4-s + 5-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)7-s + (0.809 − 0.587i)8-s + 9-s + (−0.309 + 0.951i)10-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s − 14-s + 15-s + (0.309 + 0.951i)16-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + 3-s + (−0.809 − 0.587i)4-s + 5-s + (−0.309 + 0.951i)6-s + (0.309 + 0.951i)7-s + (0.809 − 0.587i)8-s + 9-s + (−0.309 + 0.951i)10-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s − 14-s + 15-s + (0.309 + 0.951i)16-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.775 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.926731481 - 0.6858848538i\)
\(L(\frac12)\) \(\approx\) \(1.926731481 - 0.6858848538i\)
\(L(1)\) \(\approx\) \(1.237999907 + 0.3897764885i\)
\(L(1)\) \(\approx\) \(1.237999907 + 0.3897764885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 - T \)
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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

−20.45030492521084462466172957016, −20.07906502319562347841489708478, −19.28222509080243344259529010053, −18.36129455868142926158065039659, −17.95509877557578623117717304469, −16.85588802282944290420624985994, −16.504539042586233355560286063219, −14.97492548158437778981554116241, −14.14337597253346405637673387467, −13.82595346773170400406869317272, −13.10048430348879475377056402544, −12.31087116280816689654302229132, −11.26141056812876141667778177169, −10.38724522131242989048688536573, −9.72761095717420763642962655658, −9.30382446688336222120474333333, −8.338079563371649912467667323639, −7.492204399927968627589418293123, −6.79713521200601889977274868717, −5.20937099895913470871538411967, −4.474997016120559934972713245361, −3.54882738775803629097240715546, −2.72571339882416065220855603058, −1.742377876056215608215087052554, −1.311736767223960819666926601693, 0.3285656089473171225918042214, 1.915294529578081432047739577, 2.26085906991946477937485569891, 3.63699742323715640492574536366, 4.82003847611066312126568717938, 5.426360707946378940176978718226, 6.4106989634442686216545555751, 7.11385951681917732138600113861, 8.2402697384061614692574236898, 8.62346748701179563984814834026, 9.43661517649630721324784784421, 10.02514680721158468359938840622, 10.88374898892907557797343073415, 12.537370311035637426458611371595, 12.96798185506980669946254887528, 13.86116302527139847838185650989, 14.56600983095791203412153389730, 15.11927374053384665812892790789, 15.63176766661195183354109445357, 16.80582026276920880436543644985, 17.44314519940492557201420991644, 18.21065228792896080204417745615, 18.73334398087958962582692318934, 19.60847942479752641070746206676, 20.310984710699745234215523200508

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