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Properties

Label 1-283-283.264-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.0414 - 0.999i$
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  + (0.231 + 0.972i)2-s + (0.100 + 0.994i)3-s + (−0.892 + 0.451i)4-s + (−0.979 − 0.199i)5-s + (−0.944 + 0.328i)6-s + (0.920 + 0.390i)7-s + (−0.645 − 0.763i)8-s + (−0.979 + 0.199i)9-s + (−0.0334 − 0.999i)10-s + (−0.892 − 0.451i)11-s + (−0.538 − 0.842i)12-s + (−0.944 + 0.328i)13-s + (−0.166 + 0.986i)14-s + (0.100 − 0.994i)15-s + (0.593 − 0.805i)16-s + (−0.166 − 0.986i)17-s + ⋯
L(s)  = 1  + (0.231 + 0.972i)2-s + (0.100 + 0.994i)3-s + (−0.892 + 0.451i)4-s + (−0.979 − 0.199i)5-s + (−0.944 + 0.328i)6-s + (0.920 + 0.390i)7-s + (−0.645 − 0.763i)8-s + (−0.979 + 0.199i)9-s + (−0.0334 − 0.999i)10-s + (−0.892 − 0.451i)11-s + (−0.538 − 0.842i)12-s + (−0.944 + 0.328i)13-s + (−0.166 + 0.986i)14-s + (0.100 − 0.994i)15-s + (0.593 − 0.805i)16-s + (−0.166 − 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0414 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0414 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.0414 - 0.999i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.0414 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2139581653 + 0.2230134876i\)
\(L(\frac12)\) \(\approx\) \(-0.2139581653 + 0.2230134876i\)
\(L(1)\) \(\approx\) \(0.4273797608 + 0.5410195849i\)
\(L(1)\) \(\approx\) \(0.4273797608 + 0.5410195849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.231 + 0.972i)T \)
3 \( 1 + (0.100 + 0.994i)T \)
5 \( 1 + (-0.979 - 0.199i)T \)
7 \( 1 + (0.920 + 0.390i)T \)
11 \( 1 + (-0.892 - 0.451i)T \)
13 \( 1 + (-0.944 + 0.328i)T \)
17 \( 1 + (-0.166 - 0.986i)T \)
19 \( 1 + (-0.944 + 0.328i)T \)
23 \( 1 + (-0.824 + 0.565i)T \)
29 \( 1 + (-0.420 + 0.907i)T \)
31 \( 1 + (0.964 - 0.264i)T \)
37 \( 1 + (0.359 - 0.933i)T \)
41 \( 1 + (-0.645 + 0.763i)T \)
43 \( 1 + (-0.741 - 0.670i)T \)
47 \( 1 + (-0.944 - 0.328i)T \)
53 \( 1 + (0.593 + 0.805i)T \)
59 \( 1 + (0.695 + 0.718i)T \)
61 \( 1 + (-0.0334 + 0.999i)T \)
67 \( 1 + (-0.538 + 0.842i)T \)
71 \( 1 + (-0.892 - 0.451i)T \)
73 \( 1 + (0.964 + 0.264i)T \)
79 \( 1 + (-0.741 + 0.670i)T \)
83 \( 1 + (-0.296 - 0.955i)T \)
89 \( 1 + (-0.741 - 0.670i)T \)
97 \( 1 + (-0.824 + 0.565i)T \)
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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

−24.42560606013251013766224154246, −23.84371130147359394324292327273, −23.2282622696291588201145549042, −22.31197555936965342547070926028, −21.06306895189615174705902224871, −20.18340810260210289884242220893, −19.52795388813510568965179660933, −18.74316729295499821565138817757, −17.830341672381265369038189094694, −17.109053136395036950357456163708, −15.14186761413100786703978027715, −14.66025192828939363809316628533, −13.464935713826922349606488376991, −12.607063985343561194435651956913, −11.87319148122742879793200988151, −10.99415460805786345060899943455, −10.09971209377916725726894101391, −8.19717057510305074253299768944, −8.09257662376999862109132287924, −6.63156240571392773366427051370, −5.08312653160752701096006733355, −4.151185326275978375874442404370, −2.7453202030403290698864897599, −1.81341362755435770438142482260, −0.182081081347283000633629481117, 2.77556429423094345861520231941, 4.096190771262095233062118779021, 4.84410991783048062234270408099, 5.61054334804764926737445837017, 7.285770709988951505752558917435, 8.19051865710615723120209439718, 8.82874968565054260761616431887, 10.06947821731263450805635059382, 11.36403351026276939385072128789, 12.168184132823028329536817932361, 13.586664030848921328240046064704, 14.67865854711544717795207394272, 15.15950390536399242270646357896, 16.08152992219903236155602629764, 16.68180094880958151871057545857, 17.820089055943115229753874882436, 18.825128056606166824791483125387, 20.02545256081307847075602421436, 21.13283301422002849545875564209, 21.736338781035604002975125237050, 22.769570984634674565837046499304, 23.644145699443529677891365836067, 24.33459385336899426128240346723, 25.32794267678162066334305534133, 26.46145215624390213349822602959

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