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Properties

Label 1-2280-2280.1259-r0-0-0
Degree $1$
Conductor $2280$
Sign $-0.378 + 0.925i$
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  + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)29-s + (0.5 − 0.866i)31-s + 37-s + (0.939 + 0.342i)41-s + (−0.173 − 0.984i)43-s + (−0.766 − 0.642i)47-s + (−0.5 − 0.866i)49-s + (−0.173 + 0.984i)53-s + (−0.766 + 0.642i)59-s + (−0.173 + 0.984i)61-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)29-s + (0.5 − 0.866i)31-s + 37-s + (0.939 + 0.342i)41-s + (−0.173 − 0.984i)43-s + (−0.766 − 0.642i)47-s + (−0.5 − 0.866i)49-s + (−0.173 + 0.984i)53-s + (−0.766 + 0.642i)59-s + (−0.173 + 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.378 + 0.925i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ -0.378 + 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6789725887 + 1.011046419i\)
\(L(\frac12)\) \(\approx\) \(0.6789725887 + 1.011046419i\)
\(L(1)\) \(\approx\) \(0.9411933621 + 0.2700992316i\)
\(L(1)\) \(\approx\) \(0.9411933621 + 0.2700992316i\)

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 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−19.455920519594826772497606891410, −18.96933271135914712526309236599, −17.84808542021919511803278432022, −17.247964105500890034356929234647, −16.488594554761088002036102982522, −16.1263758008093444980442707849, −14.96369179539151519824018039863, −14.32935177039869062646586923930, −13.76529785235489198493852342203, −12.79691546836901308573567183880, −12.323944519893164932970415720360, −11.34194662139613289892566100128, −10.56213198033521220651687165459, −9.956805632445518242991863333834, −9.238848334191659100761425567760, −8.12642338999253604972514158028, −7.72058524591574030725906601099, −6.50813407419419256121756090790, −6.229773760486813571053663251123, −5.009894069162658766954620178659, −4.246899540935977010177928473711, −3.35780223008966681272558326688, −2.68269731678838240498223069887, −1.34871823183223924220194579113, −0.44185061569471021493699525232, 1.18421473490137683452633063930, 2.27980376228881970881896907795, 2.89999731194367132862118670383, 3.97641770778536476022352019596, 4.853763823392871005152732141295, 5.59738697382742288106264644142, 6.464995743400200131356070879213, 7.24660374143889543176005005253, 7.937339822172829271239012432658, 9.06776285420959008304605671543, 9.57632483881023428004096439025, 10.09069354959630602700010088139, 11.31190535598344052248951168900, 12.12209610540663134249489372790, 12.3138842188464826649134429441, 13.36349423444581118756831959486, 14.17726777579299462580453896447, 14.94914535868316182888130147599, 15.430186469979657287182679276574, 16.37196011926780593488857015670, 16.92967022153491934213722737896, 17.83676314628573628514911519467, 18.39845605754106676901944986359, 19.28360529837243424960194651564, 19.735074342563830820970245179119

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