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Properties

Label 1-283-283.64-r0-0-0
Degree $1$
Conductor $283$
Sign $-0.0427 + 0.999i$
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  + (0.784 − 0.619i)2-s + (−0.741 + 0.670i)3-s + (0.231 − 0.972i)4-s + (0.100 + 0.994i)5-s + (−0.166 + 0.986i)6-s + (−0.979 + 0.199i)7-s + (−0.420 − 0.907i)8-s + (0.100 − 0.994i)9-s + (0.695 + 0.718i)10-s + (0.231 + 0.972i)11-s + (0.480 + 0.876i)12-s + (−0.166 + 0.986i)13-s + (−0.645 + 0.763i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (−0.645 − 0.763i)17-s + ⋯
L(s)  = 1  + (0.784 − 0.619i)2-s + (−0.741 + 0.670i)3-s + (0.231 − 0.972i)4-s + (0.100 + 0.994i)5-s + (−0.166 + 0.986i)6-s + (−0.979 + 0.199i)7-s + (−0.420 − 0.907i)8-s + (0.100 − 0.994i)9-s + (0.695 + 0.718i)10-s + (0.231 + 0.972i)11-s + (0.480 + 0.876i)12-s + (−0.166 + 0.986i)13-s + (−0.645 + 0.763i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (−0.645 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0427 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0427 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.0427 + 0.999i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ -0.0427 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6543093178 + 0.6828957163i\)
\(L(\frac12)\) \(\approx\) \(0.6543093178 + 0.6828957163i\)
\(L(1)\) \(\approx\) \(0.9906126539 + 0.1774007776i\)
\(L(1)\) \(\approx\) \(0.9906126539 + 0.1774007776i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (0.784 - 0.619i)T \)
3 \( 1 + (-0.741 + 0.670i)T \)
5 \( 1 + (0.100 + 0.994i)T \)
7 \( 1 + (-0.979 + 0.199i)T \)
11 \( 1 + (0.231 + 0.972i)T \)
13 \( 1 + (-0.166 + 0.986i)T \)
17 \( 1 + (-0.645 - 0.763i)T \)
19 \( 1 + (-0.166 + 0.986i)T \)
23 \( 1 + (-0.296 + 0.955i)T \)
29 \( 1 + (-0.538 + 0.842i)T \)
31 \( 1 + (0.991 + 0.133i)T \)
37 \( 1 + (-0.824 - 0.565i)T \)
41 \( 1 + (-0.420 + 0.907i)T \)
43 \( 1 + (0.359 + 0.933i)T \)
47 \( 1 + (-0.166 - 0.986i)T \)
53 \( 1 + (-0.892 + 0.451i)T \)
59 \( 1 + (0.920 - 0.390i)T \)
61 \( 1 + (0.695 - 0.718i)T \)
67 \( 1 + (0.480 - 0.876i)T \)
71 \( 1 + (0.231 + 0.972i)T \)
73 \( 1 + (0.991 - 0.133i)T \)
79 \( 1 + (0.359 - 0.933i)T \)
83 \( 1 + (0.593 + 0.805i)T \)
89 \( 1 + (0.359 + 0.933i)T \)
97 \( 1 + (-0.296 + 0.955i)T \)
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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

−25.02126403532466108507045564144, −24.170807972289081327967569817907, −23.8801544684703849876293809322, −22.450596828826837034177680583138, −22.32204367385844098287621443294, −21.01550364224491552328712909969, −19.91054103481632544869187358730, −19.02928534331596714083419886684, −17.44744180750711759975276144735, −17.12942189920711734571174289777, −16.12793612332941254464658647258, −15.48926147665488338394738886694, −13.80507402337197875590311634343, −13.15763105120671073123022489362, −12.60589038994700463916025787772, −11.65706993198172952777512164403, −10.455873856282302042789914554441, −8.80754353680452043801875821132, −7.97887533962871048651820287792, −6.67644894130940703826458005779, −6.012379775374303353906325542879, −5.09595015177228473161739434817, −3.93055157577407097224557615003, −2.46074173243110973334869570283, −0.5096170384093255626593527199, 1.9496458065604988372981958499, 3.270520750175111644487349431622, 4.11569704480772197445846612945, 5.28878446984017970888567173605, 6.45556581726432198472495541433, 6.91809233879713938615455445115, 9.52128434321037728329282566969, 9.79476165469158747041025312843, 10.90389377312752702050769724651, 11.7562173524937808842462195749, 12.5020223314056973823128456235, 13.78291469602184390734327717653, 14.72719017700495670620284522024, 15.532976592882074592511864995321, 16.34877747665383135986664800839, 17.7197659303824255374587244516, 18.638606914472675883338845214677, 19.53816629143327238238930799058, 20.58808328575654861967144402959, 21.640843955003998285181768197656, 22.17339955974961084648400534037, 22.96563668311965419143819087124, 23.362390973631216099859389299101, 24.822507715228783072566109237295, 25.89889148239803217805054666807

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