Properties

Label 1-2280-2280.2099-r0-0-0
Degree $1$
Conductor $2280$
Sign $0.486 + 0.873i$
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 + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)29-s + (0.5 + 0.866i)31-s + 37-s + (−0.173 + 0.984i)41-s + (−0.766 − 0.642i)43-s + (0.939 + 0.342i)47-s + (−0.5 + 0.866i)49-s + (−0.766 + 0.642i)53-s + (0.939 − 0.342i)59-s + (−0.766 + 0.642i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)17-s + (−0.766 + 0.642i)23-s + (−0.939 − 0.342i)29-s + (0.5 + 0.866i)31-s + 37-s + (−0.173 + 0.984i)41-s + (−0.766 − 0.642i)43-s + (0.939 + 0.342i)47-s + (−0.5 + 0.866i)49-s + (−0.766 + 0.642i)53-s + (0.939 − 0.342i)59-s + (−0.766 + 0.642i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (2099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ 0.486 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9369995324 + 0.5508956363i\)
\(L(\frac12)\) \(\approx\) \(0.9369995324 + 0.5508956363i\)
\(L(1)\) \(\approx\) \(0.9323125819 + 0.02353623058i\)
\(L(1)\) \(\approx\) \(0.9323125819 + 0.02353623058i\)

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

−19.67031002060727691231459941206, −18.67294659508480872161379156967, −18.16709497283983687046546166227, −17.48654959330707941821351622592, −16.64970628629264858349638430464, −15.81920087551846359319917354032, −15.2151277447065733599820906640, −14.71625030181254879089127993862, −13.64051733372629018956084644326, −12.91008086376462170221149391068, −12.343057373067697575944078015055, −11.60247314926394794521729499807, −10.76862397542376658754620976136, −9.83393401612379478766439259262, −9.32678437392936244783313190452, −8.48435112417183408463839308238, −7.70659152709733236626705556124, −6.73478779058143561932247199769, −6.10957932996468527138253032919, −5.27347330552323369994674573777, −4.39599835957857173250141296254, −3.504685994013459801994313940979, −2.51394078240832827049328203963, −1.88911862428535968011344061255, −0.40443623385513619869602253169, 0.97887742906511695241656126299, 1.9076038300994443227177771848, 3.06180690438723617637938069049, 3.94197053417943808301057002927, 4.382943874752514537416193354978, 5.66980523348926519020599638225, 6.432176770638256318873929612634, 6.972696706530630935056477936388, 7.9432286450876320786193607327, 8.768673300960211964775877291, 9.48936897941654088762537391952, 10.21827529255022910650999889255, 11.19380334333715046960013490205, 11.52106753438555660889098399086, 12.61800646910755929919501382485, 13.435655975931323852076860698786, 13.87138951004528185225269914249, 14.59108700686743421204576063837, 15.65226157060287065633891594347, 16.20320970781771574284726096375, 16.95357456581919259150805368557, 17.41437580007998616946262599415, 18.51233574172670594168339360855, 19.05760907961501589032446014862, 19.91186990558033859663996269924

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