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Properties

Label 1-1045-1045.7-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.0617 + 0.998i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (−0.207 + 0.978i)2-s + (−0.406 − 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.669 + 0.743i)9-s + i·12-s + (0.743 + 0.669i)13-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (−0.587 − 0.809i)18-s + (0.5 − 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.978 − 0.207i)24-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.406 − 0.913i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.669 + 0.743i)9-s + i·12-s + (0.743 + 0.669i)13-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (−0.587 − 0.809i)18-s + (0.5 − 0.866i)21-s + (−0.866 + 0.5i)23-s + (−0.978 − 0.207i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0617 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0617 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.0617 + 0.998i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.0617 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6854499709 + 0.7291378909i\)
\(L(\frac12)\) \(\approx\) \(0.6854499709 + 0.7291378909i\)
\(L(1)\) \(\approx\) \(0.7724947958 + 0.2997365268i\)
\(L(1)\) \(\approx\) \(0.7724947958 + 0.2997365268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.207 + 0.978i)T \)
3 \( 1 + (-0.406 - 0.913i)T \)
7 \( 1 + (0.587 + 0.809i)T \)
13 \( 1 + (0.743 + 0.669i)T \)
17 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.913 + 0.406i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.994 - 0.104i)T \)
53 \( 1 + (-0.743 - 0.669i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.978 - 0.207i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.994 - 0.104i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.207 - 0.978i)T \)
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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

−21.2459983224035045521840184287, −20.48392600600460303378259116094, −20.21137056977762292175597177692, −19.08421470648870907777652536447, −18.16608830854057022526889162687, −17.41956144408291849710258910174, −16.91555151767608749389317687238, −16.00558284566278270346369670900, −14.97063975946627682386831106624, −14.16494358608150996265595885593, −13.404264694481036376988298045982, −12.362812042609942482577849497286, −11.577065691276260625812508015171, −10.8772143604618924001791231846, −10.18165258571819294990264199615, −9.75212096846125262369812238573, −8.296155732783204313645514332, −8.15493931577401493468986097906, −6.48482419689510995542179806951, −5.4072761692757330175343813738, −4.568021672227952789028297571040, −3.812119494738917292517845568094, −3.1050741771708538855802301898, −1.66911252522255796829647471499, −0.578802713912966152525297751962, 1.11534495332846457187906064956, 1.99218692810937018523731765752, 3.47282189551208382346923373883, 4.92537143415794496015410369484, 5.41062756238014824999103278730, 6.383663617644956931219972360371, 6.9653149364137764656447285400, 8.068834692574375257008008038973, 8.45230664074873355997907212271, 9.44593072638178350794418841581, 10.54049806568410122855512996598, 11.63056390874082011630135956091, 12.19326367468947931853616371939, 13.227314065256963055611630976, 14.061275183799619999636065856148, 14.471809507011971956024932381268, 15.733869811925435438749423918982, 16.21357397899670918561192590118, 17.16091422147733416158338513858, 17.970715173229076931401815959537, 18.35297746016438578568597078105, 19.04777528211495007904805304645, 19.87724761048537442032215275078, 21.19945118152193573989974894573, 21.89534741871994886799165240936

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