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Properties

Label 1-1045-1045.293-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.742 + 0.670i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (−0.743 − 0.669i)2-s + (0.994 + 0.104i)3-s + (0.104 + 0.994i)4-s + (−0.669 − 0.743i)6-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (0.978 + 0.207i)9-s + i·12-s + (0.207 − 0.978i)13-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.207 − 0.978i)17-s + (−0.587 − 0.809i)18-s + (−0.5 − 0.866i)21-s + (−0.866 − 0.5i)23-s + (0.669 − 0.743i)24-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)2-s + (0.994 + 0.104i)3-s + (0.104 + 0.994i)4-s + (−0.669 − 0.743i)6-s + (−0.587 − 0.809i)7-s + (0.587 − 0.809i)8-s + (0.978 + 0.207i)9-s + i·12-s + (0.207 − 0.978i)13-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.207 − 0.978i)17-s + (−0.587 − 0.809i)18-s + (−0.5 − 0.866i)21-s + (−0.866 − 0.5i)23-s + (0.669 − 0.743i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.742 + 0.670i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.742 + 0.670i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1774460483 - 0.4611481457i\)
\(L(\frac12)\) \(\approx\) \(-0.1774460483 - 0.4611481457i\)
\(L(1)\) \(\approx\) \(0.7713854693 - 0.3649589738i\)
\(L(1)\) \(\approx\) \(0.7713854693 - 0.3649589738i\)

Euler product

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

−21.698631524673133619092529752553, −20.96284599640503373544712116338, −19.921043366050611260055747554670, −19.30394227819177683991245646558, −18.85208397635624392344257033979, −18.06941789084094779183965169455, −17.17311064662772145688272032349, −16.11603587495289393317217469850, −15.64240079117199962947614271362, −14.915611375482290467939178491234, −14.11551909382001294237912800806, −13.41421258530720778718229748397, −12.38627418457491541012143569422, −11.41739779741863388398056827051, −10.1378483889790936713791273039, −9.67716367642153117338918879789, −8.63101973264443529893487935912, −8.4599709861053016513845195961, −7.245072307716637154738119164421, −6.535183368157207827970285949311, −5.72481541006506223395288143322, −4.46464337220853911891821423217, −3.41092728630684669079935165100, −2.16567536928197740347051229295, −1.55722487691343768125577816382, 0.11335649046142993958092848728, 1.14398138427126559373566092511, 2.30246685704675492069021772521, 3.177589110162293679559147364676, 3.78663598890020826594279157624, 4.81436979661228174392549485119, 6.48489359467651784275705335896, 7.38540921742316087437047761926, 7.97476690179485084037745664499, 8.87895175907857102003666916923, 9.66793887604362773428941671677, 10.2711480957491074958457959813, 10.99195292680019986512983553167, 12.17978802249207962148002156802, 13.02866046607206964273208523034, 13.522788948560505614512675664289, 14.44585627104222711207945436188, 15.58741422640416723585845738503, 16.21728059720765736589357828646, 16.95634873873717463947805285423, 18.194964887743490962451814482824, 18.42418062161006450304956493795, 19.70883991072506552993184886399, 19.945603727140870081350776642636, 20.5407157468014575174215080706

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