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Properties

Label 1-1441-1441.1261-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.0555 + 0.998i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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 + (0.926 − 0.377i)3-s + (−0.607 + 0.794i)4-s + (−0.0724 − 0.997i)5-s + (−0.748 − 0.663i)6-s + (−0.399 − 0.916i)7-s + (0.981 + 0.192i)8-s + (0.715 − 0.698i)9-s + (−0.861 + 0.506i)10-s + (−0.262 + 0.964i)12-s + (−0.215 + 0.976i)13-s + (−0.644 + 0.764i)14-s + (−0.443 − 0.896i)15-s + (−0.262 − 0.964i)16-s + (−0.354 + 0.935i)17-s + (−0.943 − 0.331i)18-s + ⋯
L(s)  = 1  + (−0.443 − 0.896i)2-s + (0.926 − 0.377i)3-s + (−0.607 + 0.794i)4-s + (−0.0724 − 0.997i)5-s + (−0.748 − 0.663i)6-s + (−0.399 − 0.916i)7-s + (0.981 + 0.192i)8-s + (0.715 − 0.698i)9-s + (−0.861 + 0.506i)10-s + (−0.262 + 0.964i)12-s + (−0.215 + 0.976i)13-s + (−0.644 + 0.764i)14-s + (−0.443 − 0.896i)15-s + (−0.262 − 0.964i)16-s + (−0.354 + 0.935i)17-s + (−0.943 − 0.331i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0555 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0555 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.0555 + 0.998i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.0555 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1559976092 - 0.1475576219i\)
\(L(\frac12)\) \(\approx\) \(-0.1559976092 - 0.1475576219i\)
\(L(1)\) \(\approx\) \(0.5934891269 - 0.5229983743i\)
\(L(1)\) \(\approx\) \(0.5934891269 - 0.5229983743i\)

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 \( 1 + (0.926 - 0.377i)T \)
5 \( 1 + (-0.0724 - 0.997i)T \)
7 \( 1 + (-0.399 - 0.916i)T \)
13 \( 1 + (-0.215 + 0.976i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (-0.262 + 0.964i)T \)
23 \( 1 + (-0.981 + 0.192i)T \)
29 \( 1 + (-0.168 - 0.985i)T \)
31 \( 1 + (-0.981 - 0.192i)T \)
37 \( 1 + (-0.958 + 0.285i)T \)
41 \( 1 + (0.354 + 0.935i)T \)
43 \( 1 + (-0.644 - 0.764i)T \)
47 \( 1 + (0.443 + 0.896i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.998 - 0.0483i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.644 + 0.764i)T \)
71 \( 1 + (-0.981 - 0.192i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.0724 - 0.997i)T \)
83 \( 1 + (0.958 + 0.285i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.399 - 0.916i)T \)
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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

−21.52618285007456544707909615087, −20.17537184037776880137953852783, −19.71607863561760412722410827729, −18.92397296161505702620607882203, −18.210878096580458470943999600845, −17.84546105528493577767116427838, −16.52096289127282567434374340465, −15.69183931328539010178270812209, −15.38357721482554266857113111061, −14.69253455567992989958027049443, −13.99124577782505078150209124609, −13.27156483511584473906909633453, −12.259429931954945827688040000867, −10.871649730560703005143968330799, −10.39065069613911340455351484343, −9.379130282738970913423741352592, −9.002126309717056383996017050900, −8.012968399334567527205181440788, −7.31418872859065967961931835673, −6.60053290374487971290309598300, −5.58361906689095636305612313413, −4.79502804746333658446787752242, −3.57590819025095674297828020290, −2.77334818587783527204428697399, −1.918792618630270417064497471223, 0.07679703338221355342219141127, 1.587099932292776761200397914208, 1.78034778231721656473831148881, 3.2035811588723336133489467144, 4.13662057984001462839790524180, 4.32524762429684988345564208450, 6.01722233342792318672936433174, 7.159260033830296022426223292436, 7.95851589019271415400660777012, 8.49442681239957619947690958572, 9.43146285456448260728365412866, 9.84304837101850846611079679166, 10.80589581319597106605656654586, 11.95540403138605170056648190871, 12.48090248369498165649559269511, 13.25785684949148540556404979842, 13.749282757261025801804918008562, 14.55063824565508209048642009104, 15.8008300942682146936792330021, 16.6267386658836477071427540626, 17.1614881047118055930478632550, 18.03584701674454671742625249203, 19.16800110568296876130770507613, 19.30328448091595324510193424803, 20.23994427495651407727013498078

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