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Properties

Label 1-283-283.66-r0-0-0
Degree $1$
Conductor $283$
Sign $0.564 - 0.825i$
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  + (0.860 − 0.509i)2-s + (0.784 + 0.619i)3-s + (0.480 − 0.876i)4-s + (0.231 − 0.972i)5-s + (0.991 + 0.133i)6-s + (−0.892 − 0.451i)7-s + (−0.0334 − 0.999i)8-s + (0.231 + 0.972i)9-s + (−0.296 − 0.955i)10-s + (0.480 + 0.876i)11-s + (0.920 − 0.390i)12-s + (0.991 + 0.133i)13-s + (−0.997 + 0.0667i)14-s + (0.784 − 0.619i)15-s + (−0.538 − 0.842i)16-s + (−0.997 − 0.0667i)17-s + ⋯
L(s)  = 1  + (0.860 − 0.509i)2-s + (0.784 + 0.619i)3-s + (0.480 − 0.876i)4-s + (0.231 − 0.972i)5-s + (0.991 + 0.133i)6-s + (−0.892 − 0.451i)7-s + (−0.0334 − 0.999i)8-s + (0.231 + 0.972i)9-s + (−0.296 − 0.955i)10-s + (0.480 + 0.876i)11-s + (0.920 − 0.390i)12-s + (0.991 + 0.133i)13-s + (−0.997 + 0.0667i)14-s + (0.784 − 0.619i)15-s + (−0.538 − 0.842i)16-s + (−0.997 − 0.0667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.240286430 - 1.182721429i\)
\(L(\frac12)\) \(\approx\) \(2.240286430 - 1.182721429i\)
\(L(1)\) \(\approx\) \(1.931195190 - 0.6440799166i\)
\(L(1)\) \(\approx\) \(1.931195190 - 0.6440799166i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.860 - 0.509i)T \)
3 \( 1 + (0.784 + 0.619i)T \)
5 \( 1 + (0.231 - 0.972i)T \)
7 \( 1 + (-0.892 - 0.451i)T \)
11 \( 1 + (0.480 + 0.876i)T \)
13 \( 1 + (0.991 + 0.133i)T \)
17 \( 1 + (-0.997 - 0.0667i)T \)
19 \( 1 + (0.991 + 0.133i)T \)
23 \( 1 + (-0.645 - 0.763i)T \)
29 \( 1 + (0.695 - 0.718i)T \)
31 \( 1 + (-0.741 - 0.670i)T \)
37 \( 1 + (-0.166 + 0.986i)T \)
41 \( 1 + (-0.0334 + 0.999i)T \)
43 \( 1 + (-0.944 - 0.328i)T \)
47 \( 1 + (0.991 - 0.133i)T \)
53 \( 1 + (-0.538 + 0.842i)T \)
59 \( 1 + (0.593 + 0.805i)T \)
61 \( 1 + (-0.296 + 0.955i)T \)
67 \( 1 + (0.920 + 0.390i)T \)
71 \( 1 + (0.480 + 0.876i)T \)
73 \( 1 + (-0.741 + 0.670i)T \)
79 \( 1 + (-0.944 + 0.328i)T \)
83 \( 1 + (-0.420 + 0.907i)T \)
89 \( 1 + (-0.944 - 0.328i)T \)
97 \( 1 + (-0.645 - 0.763i)T \)
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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

−25.546503249307819398879399912865, −24.94206657719819210055126647650, −23.95158048246953025311292603400, −23.06740978411978885231449076459, −22.08191583424749044840190117072, −21.571666254239529265633138791678, −20.22920891729148744779726835279, −19.435262007258120472059650629357, −18.380040753079905333536107517702, −17.65490366281925646415119054473, −16.00999339212547790438394156304, −15.565084782512637706167837342119, −14.329325184664524052175318883396, −13.78747488199347048517292641633, −13.03693971031995748368928266529, −11.93017002251318421822162459546, −10.92043595243446683761008979411, −9.31185056662636050572341133601, −8.4222100221705328561127256961, −7.15708582695512447418726057172, −6.47348879741064068466607122840, −5.70151956618288914237830787279, −3.576401427372551873707703423287, −3.27227292173805455525748911215, −2.00855103755733576261118317775, 1.419382877600195175751148030556, 2.681383037838368778379871751642, 3.99858766555690650358842308092, 4.44383226751770567090922589637, 5.8026455083926593155876534973, 6.99051475743123983417293097102, 8.55435636113338278428455025312, 9.60770311098756962600428076953, 10.11537493032779990993264674380, 11.46868920395586793261129926329, 12.628663451309058321355791129042, 13.44721495546137122597760951398, 13.982819928692610643951434227682, 15.3141582145771484929762470597, 15.956988193059385967587428303865, 16.79907894368720757188385191739, 18.416200606133689302000871123471, 19.6716354376341523656964407802, 20.2892445527109438904375623228, 20.607341065902329800126529747383, 21.84898428295354300760778599170, 22.498923161883973897524229575361, 23.51164822301665134298836709710, 24.61816187714575204086711775132, 25.273216444955551806641594814436

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