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Properties

Label 1-2280-2280.1973-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.654 + 0.756i$
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  + (0.866 + 0.5i)7-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.984 + 0.173i)17-s + (0.342 + 0.939i)23-s + (−0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.984 + 0.173i)47-s + (0.5 + 0.866i)49-s + (−0.342 − 0.939i)53-s + (−0.173 − 0.984i)59-s + (0.939 − 0.342i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (−0.5 − 0.866i)11-s + (0.642 − 0.766i)13-s + (−0.984 + 0.173i)17-s + (0.342 + 0.939i)23-s + (−0.173 + 0.984i)29-s + (−0.5 + 0.866i)31-s − i·37-s + (−0.766 + 0.642i)41-s + (−0.342 + 0.939i)43-s + (0.984 + 0.173i)47-s + (0.5 + 0.866i)49-s + (−0.342 − 0.939i)53-s + (−0.173 − 0.984i)59-s + (0.939 − 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.654 + 0.756i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.654 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.467223699 + 0.6709844289i\)
\(L(\frac12)\) \(\approx\) \(1.467223699 + 0.6709844289i\)
\(L(1)\) \(\approx\) \(1.121755486 + 0.1163281059i\)
\(L(1)\) \(\approx\) \(1.121755486 + 0.1163281059i\)

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

−19.678053429544146966324822176522, −18.546006368750458791242329991294, −18.31762145246617931684503800711, −17.259252784323257076801023568092, −16.94357324816268733844092746499, −15.83737241508140980583388588487, −15.28176862122812513820733830632, −14.49619138549734040535850194188, −13.749670949948504107597812642567, −13.16449764817395600994814455727, −12.24715063603700336475141356196, −11.444280212273567360177825141247, −10.82535249959737350419459800053, −10.15572866319347190914021370909, −9.12323829764150952702036620879, −8.545680480219080982596249103028, −7.5575761906060538033268740284, −7.04771801934622123676127200864, −6.114145442370328170492495848638, −5.11216179979645166598009263767, −4.36882278007255141963009983959, −3.83062650911232761591738163170, −2.32462628107606015828237622610, −1.92106720714729107812316294719, −0.59417298928575489630697672677, 1.03331266194465050010136365633, 1.913282426999945704229182624469, 2.977667685261810740009962839328, 3.6435303115903160702576545643, 4.90884345162859695794539951307, 5.34436520144804131834949650164, 6.225319016778412077180286398647, 7.11361307583811127567491312321, 8.27664149641372875653808377504, 8.37534781852305717962559620700, 9.357044032503111764635007631281, 10.388229737842439077979619821711, 11.19584910085236358756744324072, 11.41219430076600656582834115420, 12.65778546041156509348787167969, 13.17725878373774469991537984006, 13.99011449926310389025063909680, 14.72786333364410123185880783839, 15.5892849226910471693965386949, 15.90632052506159746930427618466, 17.01960712454831838062162424354, 17.67781074006828365758557122009, 18.35275231416943246032152413863, 18.82672579565178185599834067375, 19.92756447147081673415644598937

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