Properties

Label 1-2280-2280.1733-r0-0-0
Degree $1$
Conductor $2280$
Sign $-0.808 - 0.588i$
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 + (−0.342 + 0.939i)13-s + (−0.642 + 0.766i)17-s + (0.984 − 0.173i)23-s + (−0.766 + 0.642i)29-s + (−0.5 − 0.866i)31-s i·37-s + (0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (0.642 + 0.766i)47-s + (0.5 − 0.866i)49-s + (−0.984 + 0.173i)53-s + (−0.766 − 0.642i)59-s + (−0.173 − 0.984i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (−0.642 + 0.766i)17-s + (0.984 − 0.173i)23-s + (−0.766 + 0.642i)29-s + (−0.5 − 0.866i)31-s i·37-s + (0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (0.642 + 0.766i)47-s + (0.5 − 0.866i)49-s + (−0.984 + 0.173i)53-s + (−0.766 − 0.642i)59-s + (−0.173 − 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.808 - 0.588i$
Analytic conductor: \(10.5882\)
Root analytic conductor: \(10.5882\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2280} (1733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2280,\ (0:\ ),\ -0.808 - 0.588i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05341980155 + 0.1641677346i\)
\(L(\frac12)\) \(\approx\) \(-0.05341980155 + 0.1641677346i\)
\(L(1)\) \(\approx\) \(0.7280325075 + 0.1779429939i\)
\(L(1)\) \(\approx\) \(0.7280325075 + 0.1779429939i\)

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

−19.35043211791439054483168248983, −18.50049129418311404965644504758, −17.850567845483435823889982701569, −16.97329413386161170556736133206, −16.357269213591264907091159844854, −15.694117394573252356101447032955, −15.02023533056170748846728151721, −14.031579874711686892027787298678, −13.30466384702597587456173074602, −12.91685201674748194527408010946, −11.99887413445840896445824368230, −10.91529047556112045441949231227, −10.64442412387860891424948439843, −9.56491249478352276181972757706, −9.054599326436049977760648068347, −8.00209618877570045000729136791, −7.32287302559547736066002611314, −6.56087924969793904865370863841, −5.65414841957868145907118414075, −4.99637393648079826031040337252, −3.86774606582370708910414096379, −3.1160238356245936488142248728, −2.46059847334388679173318500506, −0.99505722939997441763692178557, −0.06250001741242112982156835037, 1.63408463173092916756190018339, 2.37001469045806369790128608203, 3.262645366347535419558622058, 4.2480337809734095589005506578, 4.98285654947360067878372530277, 5.94619825681254103329855102273, 6.71219807057840517007471023284, 7.323665546111699773565743122932, 8.33207862572207664320869621874, 9.27483037341939684577304756080, 9.58560952334637950702476416439, 10.61248763902152224148987530395, 11.30055455091451787553384109580, 12.28686456026932810358866218977, 12.79863210875307098957556480636, 13.3905155288453491063214153376, 14.45321607555322418756293279071, 15.14197032748226316753023608223, 15.661368233467384990806375575428, 16.57563075669558922929746523592, 17.125052371273525799775240758435, 17.97367673071828320358339524499, 18.928158095937140308758786980781, 19.07648193589915002930035856671, 20.19502874757691320365230197746

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