Properties

Label 1-2280-2280.107-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.997 - 0.0678i$
Analytic cond. $10.5882$
Root an. cond. $10.5882$
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  i·7-s − 11-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)17-s + (−0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + 31-s i·37-s + (−0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s − 49-s + (0.866 − 0.5i)53-s + (0.5 − 0.866i)59-s + (0.5 + 0.866i)61-s + ⋯
L(s)  = 1  i·7-s − 11-s + (−0.866 + 0.5i)13-s + (0.866 + 0.5i)17-s + (−0.866 + 0.5i)23-s + (−0.5 − 0.866i)29-s + 31-s i·37-s + (−0.5 + 0.866i)41-s + (0.866 + 0.5i)43-s + (0.866 − 0.5i)47-s − 49-s + (0.866 − 0.5i)53-s + (0.5 − 0.866i)59-s + (0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.997 - 0.0678i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.997 - 0.0678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.308039290 - 0.04443595656i\)
\(L(\frac12)\) \(\approx\) \(1.308039290 - 0.04443595656i\)
\(L(1)\) \(\approx\) \(0.9719094975 - 0.05008466993i\)
\(L(1)\) \(\approx\) \(0.9719094975 - 0.05008466993i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 - T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−19.61074965129557401377865929761, −18.80528387289557052147806554263, −18.32914976456017835898681147979, −17.63384220999289905830554123011, −16.77520766498849598922442271994, −15.89597012113405186510791923406, −15.489436742655532058275325853757, −14.60701548813330968383824987118, −14.01301735821285817434057806255, −12.98589332767277073389538028099, −12.30359973483718785333973953635, −11.9250188154841165428638292892, −10.75489783992673408263816742985, −10.1783445463727872356682226423, −9.368030187141103888766032084508, −8.5780375495894692366443287611, −7.77387164449447381423300269229, −7.18180244845734582216730989068, −5.94979637616247361074636725761, −5.43220528737302131489135352821, −4.750375894262622239337621629, −3.554898491030004532887262954883, −2.615282342648416110698499686523, −2.15322538106452152006622281924, −0.64196330089746958015500267521, 0.70549342775337906440558303710, 1.84187604631614782847908862013, 2.76174422380882318593975069472, 3.74154678087756491648744845209, 4.484923557265376292277509091370, 5.3082152193397977322225097752, 6.19674063972066934607498663810, 7.11340082972431518204722195224, 7.80251604756775839499097910812, 8.31046560839306819753524219358, 9.73322200919210896856596499450, 9.96439824929114259073148063793, 10.78075908633678684710807428199, 11.69003738119312892716905070811, 12.335474777617050177636767340335, 13.32785438225863297209744782600, 13.72106395738310643064486261291, 14.61569314737158872619271258252, 15.2775254005614320779223244095, 16.18869145489908404703391606927, 16.803316292319115530460168174704, 17.41404250748897475576949334171, 18.17573729853347163988894502346, 19.09872297674133868135244297818, 19.52549413062999876231738790572

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