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Properties

Label 1-283-283.28-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.204 + 0.978i$
Analytic cond. $1.31424$
Root an. cond. $1.31424$
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  + (0.964 + 0.264i)2-s + (0.756 + 0.654i)3-s + (0.860 + 0.509i)4-s + (−0.929 + 0.369i)5-s + (0.556 + 0.830i)6-s + (−0.958 − 0.285i)7-s + (0.695 + 0.718i)8-s + (0.144 + 0.989i)9-s + (−0.993 + 0.111i)10-s + (0.0111 + 0.999i)11-s + (0.317 + 0.948i)12-s + (0.441 − 0.897i)13-s + (−0.848 − 0.528i)14-s + (−0.944 − 0.328i)15-s + (0.480 + 0.876i)16-s + (0.882 + 0.470i)17-s + ⋯
L(s)  = 1  + (0.964 + 0.264i)2-s + (0.756 + 0.654i)3-s + (0.860 + 0.509i)4-s + (−0.929 + 0.369i)5-s + (0.556 + 0.830i)6-s + (−0.958 − 0.285i)7-s + (0.695 + 0.718i)8-s + (0.144 + 0.989i)9-s + (−0.993 + 0.111i)10-s + (0.0111 + 0.999i)11-s + (0.317 + 0.948i)12-s + (0.441 − 0.897i)13-s + (−0.848 − 0.528i)14-s + (−0.944 − 0.328i)15-s + (0.480 + 0.876i)16-s + (0.882 + 0.470i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.204 + 0.978i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.413276408 + 1.739264831i\)
\(L(\frac12)\) \(\approx\) \(1.413276408 + 1.739264831i\)
\(L(1)\) \(\approx\) \(1.590524437 + 0.9588948669i\)
\(L(1)\) \(\approx\) \(1.590524437 + 0.9588948669i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.964 + 0.264i)T \)
3 \( 1 + (0.756 + 0.654i)T \)
5 \( 1 + (-0.929 + 0.369i)T \)
7 \( 1 + (-0.958 - 0.285i)T \)
11 \( 1 + (0.0111 + 0.999i)T \)
13 \( 1 + (0.441 - 0.897i)T \)
17 \( 1 + (0.882 + 0.470i)T \)
19 \( 1 + (-0.997 + 0.0667i)T \)
23 \( 1 + (-0.575 + 0.818i)T \)
29 \( 1 + (0.920 + 0.390i)T \)
31 \( 1 + (0.628 - 0.777i)T \)
37 \( 1 + (-0.338 - 0.940i)T \)
41 \( 1 + (-0.970 - 0.242i)T \)
43 \( 1 + (-0.166 - 0.986i)T \)
47 \( 1 + (0.556 - 0.830i)T \)
53 \( 1 + (0.480 - 0.876i)T \)
59 \( 1 + (0.0556 - 0.998i)T \)
61 \( 1 + (0.593 - 0.805i)T \)
67 \( 1 + (-0.979 + 0.199i)T \)
71 \( 1 + (0.860 - 0.509i)T \)
73 \( 1 + (0.628 + 0.777i)T \)
79 \( 1 + (-0.166 + 0.986i)T \)
83 \( 1 + (0.999 + 0.0445i)T \)
89 \( 1 + (0.937 + 0.348i)T \)
97 \( 1 + (0.996 + 0.0890i)T \)
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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

−25.127438194615513119282761950283, −24.23415542345950032947445448335, −23.54938213371005931254992066786, −22.87155929698344506193850436664, −21.578396376064784293988643544, −20.81824631333398571446404653386, −19.80176215380687725473299240003, −19.11583178548029076375883769090, −18.70807074481714617449512003543, −16.58023970571282402585359121750, −15.96751509093440317058164982765, −15.00002792226984522035161218885, −13.97369456343382719748566806976, −13.276966465204216160595847354700, −12.22085456773017600579753722812, −11.81219777544794438539369499776, −10.36477795934686922978973656655, −8.98662680055918040712813913063, −8.08489311381792653999565576967, −6.78201141517153136849295238230, −6.12647935294508985337070605797, −4.46253669409498577805293096084, −3.46406770060120621449355540541, −2.70201289445588709136419890429, −1.07799607972893549321575017402, 2.32681609616763819469106219749, 3.56493446199064696458100365029, 3.87764576702984691067570499769, 5.21039605570277808581819200938, 6.60156713707161734797475191276, 7.5993493042932023374862131635, 8.39587633795210381309448693180, 10.04812351738725926739902746532, 10.66981948895710561364228395427, 12.092449865995657132421224817357, 12.885696013494401486217961561935, 13.89839840148041415087951371378, 14.95793817014236317013905865826, 15.45138605425908460730730907836, 16.16379819179702477248671681567, 17.23368393766274544763259853669, 18.9643777141656074771048230639, 19.816661621738879547673424097751, 20.34825551933925872630904263665, 21.40401866797371471002216411952, 22.39181188639422721211476835684, 23.04725167711432076200704083007, 23.71591998410445121009356871879, 25.29266363901883863710698750911, 25.601997045272870628438280303872

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