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Properties

Label 1-1441-1441.1025-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.140 + 0.990i$
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  + (0.168 − 0.985i)2-s + (−0.970 − 0.239i)3-s + (−0.943 − 0.331i)4-s + (−0.970 − 0.239i)5-s + (−0.399 + 0.916i)6-s + (−0.399 − 0.916i)7-s + (−0.485 + 0.873i)8-s + (0.885 + 0.464i)9-s + (−0.399 + 0.916i)10-s + (0.836 + 0.548i)12-s + (−0.215 + 0.976i)13-s + (−0.970 + 0.239i)14-s + (0.885 + 0.464i)15-s + (0.779 + 0.626i)16-s + (0.998 + 0.0483i)17-s + (0.607 − 0.794i)18-s + ⋯
L(s)  = 1  + (0.168 − 0.985i)2-s + (−0.970 − 0.239i)3-s + (−0.943 − 0.331i)4-s + (−0.970 − 0.239i)5-s + (−0.399 + 0.916i)6-s + (−0.399 − 0.916i)7-s + (−0.485 + 0.873i)8-s + (0.885 + 0.464i)9-s + (−0.399 + 0.916i)10-s + (0.836 + 0.548i)12-s + (−0.215 + 0.976i)13-s + (−0.970 + 0.239i)14-s + (0.885 + 0.464i)15-s + (0.779 + 0.626i)16-s + (0.998 + 0.0483i)17-s + (0.607 − 0.794i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.140 + 0.990i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.140 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03498708438 + 0.03038116426i\)
\(L(\frac12)\) \(\approx\) \(-0.03498708438 + 0.03038116426i\)
\(L(1)\) \(\approx\) \(0.4666309971 - 0.3536508850i\)
\(L(1)\) \(\approx\) \(0.4666309971 - 0.3536508850i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.168 - 0.985i)T \)
3 \( 1 + (-0.970 - 0.239i)T \)
5 \( 1 + (-0.970 - 0.239i)T \)
7 \( 1 + (-0.399 - 0.916i)T \)
13 \( 1 + (-0.215 + 0.976i)T \)
17 \( 1 + (0.998 + 0.0483i)T \)
19 \( 1 + (0.998 - 0.0483i)T \)
23 \( 1 + (-0.906 - 0.421i)T \)
29 \( 1 + (0.168 + 0.985i)T \)
31 \( 1 + (0.120 + 0.992i)T \)
37 \( 1 + (0.958 - 0.285i)T \)
41 \( 1 + (-0.836 - 0.548i)T \)
43 \( 1 + (-0.926 + 0.377i)T \)
47 \( 1 + (0.715 - 0.698i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.262 - 0.964i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.926 + 0.377i)T \)
71 \( 1 + (-0.681 - 0.732i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.644 - 0.764i)T \)
83 \( 1 + (-0.568 + 0.822i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.748 + 0.663i)T \)
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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

−21.53335136961620530118649707698, −20.43614316803960653666687418747, −19.3496443398249118959691580547, −18.41813246887887527204730730011, −18.2463567019557283905497802604, −17.15008618368075397326004457167, −16.45546510720810901740201624809, −15.762424157943701072239229084218, −15.325712832962806316943779186309, −14.70778792513957158308053218418, −13.48755952361321661057415371772, −12.610722892417806066064784361781, −11.97053098624566593367749306442, −11.471457721207301887329223603407, −9.990656987103612220401994959967, −9.74487100655759384294791515510, −8.382362374009405193215449801104, −7.747811973235737253305078113277, −6.99471098535082237735065470056, −5.905783045720809089386368564011, −5.6026768985339858389292082851, −4.614717429423323702065830686130, −3.71714312169471749117336211450, −2.912829816410085491661396791263, −0.89291580886214741138907538717, 0.016286623593867179120133763152, 0.87778413987907982706514049310, 1.63624655903219098986792095011, 3.16792868922077964597253186171, 3.91947607178105548387351012356, 4.65395017873131149760118669476, 5.38682984402456211550221760078, 6.58434796090531021026930392291, 7.36731437246109997926255277288, 8.219273775915568909835326186501, 9.40502963150649377137661909510, 10.16223270896754984105419755771, 10.84449255196125846296430743880, 11.62482361068250692962800176374, 12.19867973258699938943130636121, 12.69590465622659010902020295578, 13.76734589953552209912545583875, 14.29273503518150122232940578168, 15.55720057199242392600092433073, 16.55534631363921756284456992433, 16.730459401311676069375119740758, 17.93664597745644467397561055769, 18.58497831704024890719017848673, 19.247804724759463546363073123112, 19.99096465023407506396064334130

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