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Properties

Label 1-1441-1441.486-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.941 - 0.337i$
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  + (0.715 − 0.698i)2-s + (−0.0724 − 0.997i)3-s + (0.0241 − 0.999i)4-s + (0.644 + 0.764i)5-s + (−0.748 − 0.663i)6-s + (−0.215 + 0.976i)7-s + (−0.681 − 0.732i)8-s + (−0.989 + 0.144i)9-s + (0.995 + 0.0965i)10-s + (−0.998 + 0.0483i)12-s + (0.861 + 0.506i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (−0.998 − 0.0483i)16-s + (−0.354 + 0.935i)17-s + (−0.607 + 0.794i)18-s + ⋯
L(s)  = 1  + (0.715 − 0.698i)2-s + (−0.0724 − 0.997i)3-s + (0.0241 − 0.999i)4-s + (0.644 + 0.764i)5-s + (−0.748 − 0.663i)6-s + (−0.215 + 0.976i)7-s + (−0.681 − 0.732i)8-s + (−0.989 + 0.144i)9-s + (0.995 + 0.0965i)10-s + (−0.998 + 0.0483i)12-s + (0.861 + 0.506i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (−0.998 − 0.0483i)16-s + (−0.354 + 0.935i)17-s + (−0.607 + 0.794i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.941 - 0.337i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (486, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.941 - 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.089974576 - 0.3635993623i\)
\(L(\frac12)\) \(\approx\) \(2.089974576 - 0.3635993623i\)
\(L(1)\) \(\approx\) \(1.401178685 - 0.5567804258i\)
\(L(1)\) \(\approx\) \(1.401178685 - 0.5567804258i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.715 - 0.698i)T \)
3 \( 1 + (-0.0724 - 0.997i)T \)
5 \( 1 + (0.644 + 0.764i)T \)
7 \( 1 + (-0.215 + 0.976i)T \)
13 \( 1 + (0.861 + 0.506i)T \)
17 \( 1 + (-0.354 + 0.935i)T \)
19 \( 1 + (-0.998 + 0.0483i)T \)
23 \( 1 + (0.681 - 0.732i)T \)
29 \( 1 + (-0.443 + 0.896i)T \)
31 \( 1 + (0.681 + 0.732i)T \)
37 \( 1 + (0.943 + 0.331i)T \)
41 \( 1 + (0.354 + 0.935i)T \)
43 \( 1 + (0.527 - 0.849i)T \)
47 \( 1 + (-0.715 + 0.698i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.836 - 0.548i)T \)
61 \( 1 - T \)
67 \( 1 + (0.527 + 0.849i)T \)
71 \( 1 + (0.681 + 0.732i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.644 + 0.764i)T \)
83 \( 1 + (-0.943 + 0.331i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.215 + 0.976i)T \)
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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

−20.98117247291961824191411167384, −20.41273774556342763323310702389, −19.62920123377090732539018933339, −18.100233792578202953238584892430, −17.299045890990868693110006167623, −16.89955979159608615277012594546, −16.19476655900389015225121522944, −15.5582251947603680734047215830, −14.8117729224732177492875933812, −13.7514669381073980994575643378, −13.47040047184711373119098159041, −12.668512366736253752800806041551, −11.49831035466128650854871432331, −10.856787922958743162367638932927, −9.80893932968889460092050942798, −9.12421720226143969067578780262, −8.33181127679787307364973629769, −7.43149112300250875135065297570, −6.24251553971794589533293947827, −5.769966728240001541823164122295, −4.73502159834777570997663491414, −4.27030786751551863111900169289, −3.40566268020661652481504310685, −2.37202611653118603442125076823, −0.637459139238754329890854593209, 1.29509249542205406151719356841, 2.08307937790290404878707160184, 2.70300037916666859281122881844, 3.57749178330333861050413092319, 4.868977049863665717247860486559, 5.91957619453156120584734518138, 6.319643963858400963752056906087, 6.89627676863029970458629738832, 8.43779084767698729899653014794, 9.00809152229456339513215965995, 10.1118356472255477381244423088, 11.04064536065179146829054738434, 11.39917033671768928625511333922, 12.5938942714959784600953492657, 12.87977550432976927145129196343, 13.65980500734265295400630300395, 14.60199823706965889552069544355, 14.86977011540629642815272154850, 16.00931368689549233300157867008, 17.15968563892479941475222597416, 18.07584505451617241301622450834, 18.586459418349659390737550421884, 19.11270269999718089984827334941, 19.74971150007933238982160978829, 20.917524207657451827479116801761

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