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Properties

Label 1-1441-1441.312-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.328 - 0.944i$
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  + (0.861 − 0.506i)2-s + (0.779 + 0.626i)3-s + (0.485 − 0.873i)4-s + (−0.262 − 0.964i)5-s + (0.989 + 0.144i)6-s + (0.885 + 0.464i)7-s + (−0.0241 − 0.999i)8-s + (0.215 + 0.976i)9-s + (−0.715 − 0.698i)10-s + (0.926 − 0.377i)12-s + (0.885 + 0.464i)13-s + (0.998 − 0.0483i)14-s + (0.399 − 0.916i)15-s + (−0.527 − 0.849i)16-s + (−0.926 − 0.377i)17-s + (0.681 + 0.732i)18-s + ⋯
L(s)  = 1  + (0.861 − 0.506i)2-s + (0.779 + 0.626i)3-s + (0.485 − 0.873i)4-s + (−0.262 − 0.964i)5-s + (0.989 + 0.144i)6-s + (0.885 + 0.464i)7-s + (−0.0241 − 0.999i)8-s + (0.215 + 0.976i)9-s + (−0.715 − 0.698i)10-s + (0.926 − 0.377i)12-s + (0.885 + 0.464i)13-s + (0.998 − 0.0483i)14-s + (0.399 − 0.916i)15-s + (−0.527 − 0.849i)16-s + (−0.926 − 0.377i)17-s + (0.681 + 0.732i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.328 - 0.944i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (312, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.927212714 - 4.117871872i\)
\(L(\frac12)\) \(\approx\) \(2.927212714 - 4.117871872i\)
\(L(1)\) \(\approx\) \(2.119415477 - 0.9001401017i\)
\(L(1)\) \(\approx\) \(2.119415477 - 0.9001401017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.861 - 0.506i)T \)
3 \( 1 + (0.779 + 0.626i)T \)
5 \( 1 + (-0.262 - 0.964i)T \)
7 \( 1 + (0.885 + 0.464i)T \)
13 \( 1 + (0.885 + 0.464i)T \)
17 \( 1 + (-0.926 - 0.377i)T \)
19 \( 1 + (-0.644 - 0.764i)T \)
23 \( 1 + (-0.958 + 0.285i)T \)
29 \( 1 + (0.748 - 0.663i)T \)
31 \( 1 + (0.943 - 0.331i)T \)
37 \( 1 + (-0.120 - 0.992i)T \)
41 \( 1 + (-0.0724 - 0.997i)T \)
43 \( 1 + (-0.262 - 0.964i)T \)
47 \( 1 + (-0.215 - 0.976i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.926 + 0.377i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.262 - 0.964i)T \)
71 \( 1 + (0.607 + 0.794i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.998 + 0.0483i)T \)
83 \( 1 + (-0.485 - 0.873i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.443 + 0.896i)T \)
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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

−20.87497377677881720963665297588, −20.06104576222901271031163841858, −19.46846575261240301927941851172, −18.13711548171818599722963166489, −18.062279543430644489709860672057, −17.031406351538944380033862201538, −15.86667133406509371307146591868, −15.22325853604660837799854101180, −14.587018871524209185628608903030, −14.00242448632212307896983954575, −13.44141004284842434915790289370, −12.55198969901718993228665009346, −11.71453474582607054119020781612, −10.953065315284983007812484534309, −10.15759413963308788591310847811, −8.49596150581205471595748883021, −8.1933254443273067337530985017, −7.4731609420491482921906837285, −6.42840584219691803100918482729, −6.234082328793652991018953890003, −4.64819309725170301431938403788, −3.937921234036686115905553783795, −3.14394187294711490590040251753, −2.29259450923537242122485040923, −1.3244067026194777833957979889, 0.56279992001114732111854708899, 1.91917583185768464911930677493, 2.29253855068884752498708552372, 3.70046521455374721486703101150, 4.2807135778011354295331647625, 4.901084764574841971665554220184, 5.68906131017784115629234829187, 6.86161826631632878811276330027, 8.11432208318890522153201692015, 8.71507175240708874108761144436, 9.3936331068960472037127938187, 10.40575797551276447717876298674, 11.279357974822515348769193792679, 11.80072391618147929602478628977, 12.77098031980288903851170884132, 13.729060251636985683603964396123, 13.8918534103707451903797677441, 15.11832024604686217295191447077, 15.581798745828543656446008006881, 16.075299576317338155014390227, 17.20268836553718443473197289605, 18.24215907330495482976725932455, 19.211894727794698733455220317311, 19.830380664749608704078579927763, 20.45627101324754352285344212546

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