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Properties

Label 1-1441-1441.214-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.831 + 0.555i$
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  + (−0.995 + 0.0965i)2-s + (−0.262 + 0.964i)3-s + (0.981 − 0.192i)4-s + (−0.998 + 0.0483i)5-s + (0.168 − 0.985i)6-s + (0.885 − 0.464i)7-s + (−0.958 + 0.285i)8-s + (−0.861 − 0.506i)9-s + (0.989 − 0.144i)10-s + (−0.0724 + 0.997i)12-s + (0.885 − 0.464i)13-s + (−0.836 + 0.548i)14-s + (0.215 − 0.976i)15-s + (0.926 − 0.377i)16-s + (0.0724 + 0.997i)17-s + (0.906 + 0.421i)18-s + ⋯
L(s)  = 1  + (−0.995 + 0.0965i)2-s + (−0.262 + 0.964i)3-s + (0.981 − 0.192i)4-s + (−0.998 + 0.0483i)5-s + (0.168 − 0.985i)6-s + (0.885 − 0.464i)7-s + (−0.958 + 0.285i)8-s + (−0.861 − 0.506i)9-s + (0.989 − 0.144i)10-s + (−0.0724 + 0.997i)12-s + (0.885 − 0.464i)13-s + (−0.836 + 0.548i)14-s + (0.215 − 0.976i)15-s + (0.926 − 0.377i)16-s + (0.0724 + 0.997i)17-s + (0.906 + 0.421i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.831 + 0.555i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.831 + 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.185433980 + 0.3594658669i\)
\(L(\frac12)\) \(\approx\) \(1.185433980 + 0.3594658669i\)
\(L(1)\) \(\approx\) \(0.6611721860 + 0.1859851633i\)
\(L(1)\) \(\approx\) \(0.6611721860 + 0.1859851633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.995 + 0.0965i)T \)
3 \( 1 + (-0.262 + 0.964i)T \)
5 \( 1 + (-0.998 + 0.0483i)T \)
7 \( 1 + (0.885 - 0.464i)T \)
13 \( 1 + (0.885 - 0.464i)T \)
17 \( 1 + (0.0724 + 0.997i)T \)
19 \( 1 + (0.527 + 0.849i)T \)
23 \( 1 + (0.943 - 0.331i)T \)
29 \( 1 + (0.748 + 0.663i)T \)
31 \( 1 + (0.607 - 0.794i)T \)
37 \( 1 + (-0.120 + 0.992i)T \)
41 \( 1 + (0.644 - 0.764i)T \)
43 \( 1 + (-0.998 + 0.0483i)T \)
47 \( 1 + (0.861 + 0.506i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.0724 - 0.997i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.998 + 0.0483i)T \)
71 \( 1 + (-0.0241 + 0.999i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.836 - 0.548i)T \)
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

−20.18977862629590052009506310130, −19.542475756734480782058413224, −18.80902238753804677466004137211, −18.30839391185327748486597525085, −17.68533230419059519068884103212, −16.90688878700974447619214295946, −15.95568124362186382876159278291, −15.505877566069359694910068265199, −14.442855885399985532543516095210, −13.569610330277589742580403518503, −12.45883555750433599427676190201, −11.790112391341129518149326775461, −11.31630174561620769984878383920, −10.824005643808696248298814180682, −9.33010484933322698884440362851, −8.612650813536373170600673731041, −8.05569190743383190951094646726, −7.23966443313438357798424001019, −6.7266023363005859709070797112, −5.587607197344102699877838388141, −4.63834716927792385474978702299, −3.188663228922279708480538269345, −2.43604236732278007349375668797, −1.248621368465728970810924237358, −0.70893461614604150790602142018, 0.58530838450454229888817384607, 1.40738966108978583071990558444, 2.98613617623857175905662926622, 3.717977055053844012723215328929, 4.63551226773350693077292496827, 5.622456045408724346669166060649, 6.54902511794866738848229479051, 7.58924166683201742791095051837, 8.37997129716241936079605211010, 8.68117764567242331590997991005, 10.00685559522453648353737628453, 10.551309578755727397660947157841, 11.20285374716147253881968044717, 11.71726120217663908395863363396, 12.68162065614490507942149514224, 14.20251654134418678322180616086, 14.85709303377029082036738498461, 15.51577441186796291374916369679, 16.0884650278771024521294482920, 16.92151498891147400180490096810, 17.40393533500513008185049686620, 18.33988017112951704174708560918, 19.047288653014752036710111627198, 19.96167622547054620643056040522, 20.6541203227202691658307279028

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