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Properties

Label 1-1441-1441.1006-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.841 + 0.540i$
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  + (0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + 6-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)10-s + (0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + 17-s + (0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + 6-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)10-s + (0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + 17-s + (0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.841 + 0.540i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1006, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.841 + 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1375694000 + 0.4683875645i\)
\(L(\frac12)\) \(\approx\) \(0.1375694000 + 0.4683875645i\)
\(L(1)\) \(\approx\) \(0.7804818517 + 0.2127057339i\)
\(L(1)\) \(\approx\) \(0.7804818517 + 0.2127057339i\)

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 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−20.3741137190032492669661908509, −19.89704089559514144114186720184, −19.056822001491277539036232750914, −18.669420957795845679693184843632, −17.36354890466767862778771351960, −16.27280236658836212431147720067, −16.019732916518666436146620537152, −15.02267156231856207523206543490, −14.37107361462740495394582159450, −13.48192239771361040520019873774, −12.62857841236072180339227553462, −11.90380810958220527495986860560, −11.321343165331442369910749876142, −10.38441059857800521700844165998, −9.60046605789518184506700719393, −8.90317755163781935330576751204, −8.501951637588977692063725307448, −7.1182889782984980511774438353, −5.73894758660998574979313148170, −5.13402400139565771383405579446, −4.17281119559143942947133660625, −3.598387319980136702632853991354, −2.85667136860004004098623562602, −1.73977372801058073025819462728, −0.18790610892954058878037415028, 1.0299398440870368208128899180, 2.75912462241545824988187220547, 3.53540043731478262310488185671, 4.00845626177384967658253510207, 5.76088081381989290739459465595, 6.00300169180059166321834076041, 7.17259027094296502246778687611, 7.66028649944353856876974572061, 8.030185026468500967188521054473, 9.23763069851284341499945816037, 10.090090300112350674930462259255, 11.217888179159045051103384539240, 12.3095178761218318392066328126, 12.653675566774431321975023596999, 13.58787303419557591334259899073, 14.230019677909316866983659094216, 14.871776440076716971925417888734, 15.74347282742261711387666896328, 16.38767018513936934338101088632, 17.243879012802568813826022919478, 18.085511308324442115422667462310, 18.704073265317681252698420847146, 19.326115067085610453171471475970, 20.143450176053904748634932687, 20.94543704877238062505978932711

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