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Properties

Label 1-283-283.78-r0-0-0
Degree $1$
Conductor $283$
Sign $0.459 + 0.888i$
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  + (−0.892 + 0.451i)2-s + (−0.979 + 0.199i)3-s + (0.593 − 0.805i)4-s + (0.920 + 0.390i)5-s + (0.784 − 0.619i)6-s + (0.695 + 0.718i)7-s + (−0.166 + 0.986i)8-s + (0.920 − 0.390i)9-s + (−0.997 + 0.0667i)10-s + (0.593 + 0.805i)11-s + (−0.420 + 0.907i)12-s + (0.784 − 0.619i)13-s + (−0.944 − 0.328i)14-s + (−0.979 − 0.199i)15-s + (−0.296 − 0.955i)16-s + (−0.944 + 0.328i)17-s + ⋯
L(s)  = 1  + (−0.892 + 0.451i)2-s + (−0.979 + 0.199i)3-s + (0.593 − 0.805i)4-s + (0.920 + 0.390i)5-s + (0.784 − 0.619i)6-s + (0.695 + 0.718i)7-s + (−0.166 + 0.986i)8-s + (0.920 − 0.390i)9-s + (−0.997 + 0.0667i)10-s + (0.593 + 0.805i)11-s + (−0.420 + 0.907i)12-s + (0.784 − 0.619i)13-s + (−0.944 − 0.328i)14-s + (−0.979 − 0.199i)15-s + (−0.296 − 0.955i)16-s + (−0.944 + 0.328i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.459 + 0.888i$
Analytic conductor: \(1.31424\)
Root analytic conductor: \(1.31424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.459 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7118088816 + 0.4332656802i\)
\(L(\frac12)\) \(\approx\) \(0.7118088816 + 0.4332656802i\)
\(L(1)\) \(\approx\) \(0.6979660997 + 0.2597733185i\)
\(L(1)\) \(\approx\) \(0.6979660997 + 0.2597733185i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad283 \( 1 \)
good2 \( 1 + (-0.892 + 0.451i)T \)
3 \( 1 + (-0.979 + 0.199i)T \)
5 \( 1 + (0.920 + 0.390i)T \)
7 \( 1 + (0.695 + 0.718i)T \)
11 \( 1 + (0.593 + 0.805i)T \)
13 \( 1 + (0.784 - 0.619i)T \)
17 \( 1 + (-0.944 + 0.328i)T \)
19 \( 1 + (0.784 - 0.619i)T \)
23 \( 1 + (0.359 - 0.933i)T \)
29 \( 1 + (-0.645 - 0.763i)T \)
31 \( 1 + (0.860 - 0.509i)T \)
37 \( 1 + (-0.741 - 0.670i)T \)
41 \( 1 + (-0.166 - 0.986i)T \)
43 \( 1 + (0.100 + 0.994i)T \)
47 \( 1 + (0.784 + 0.619i)T \)
53 \( 1 + (-0.296 + 0.955i)T \)
59 \( 1 + (-0.0334 + 0.999i)T \)
61 \( 1 + (-0.997 - 0.0667i)T \)
67 \( 1 + (-0.420 - 0.907i)T \)
71 \( 1 + (0.593 + 0.805i)T \)
73 \( 1 + (0.860 + 0.509i)T \)
79 \( 1 + (0.100 - 0.994i)T \)
83 \( 1 + (-0.824 + 0.565i)T \)
89 \( 1 + (0.100 + 0.994i)T \)
97 \( 1 + (0.359 - 0.933i)T \)
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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

−25.40995013767224110850520701729, −24.514992129661320922664786916153, −23.89021916399903310906407829113, −22.48921281656854015947898635541, −21.59706521214167755194161298039, −20.94544751210786404056352701516, −19.99885196015801406219428027245, −18.73697862026562499780817678094, −18.02783170651658593118367989376, −17.220552427797922129219722504002, −16.68805412905428618319574549902, −15.814824963905412379092746480179, −13.89563143598505951798561051490, −13.27663161802361771221043646229, −11.92537678528412557682163839686, −11.241834870711940306280291289698, −10.45817206050492142411308322819, −9.3955361718669608011485960126, −8.424315245930339299620004210468, −7.10072184250919814999752840080, −6.2804387913587140352136190801, −5.0139840082924115069062978788, −3.664921542654032822797944625211, −1.71969182535402406811906440995, −1.10319753440019929276060627134, 1.26746186814800414921577915927, 2.38378138599837012532120359133, 4.62684443498975215501903462719, 5.699429789341035651589056747956, 6.353056055918025846562600465904, 7.37316383096215278051875833126, 8.83193648904362162191576336549, 9.60518044252751188460528481713, 10.682708099891426741026776241444, 11.26720908487076038859444264061, 12.43760200815793421467057989497, 13.847884352576139522468342108851, 15.130080027858510866214527830750, 15.541767245309849281113428210314, 16.91521818212630854193995784097, 17.641557387502511561231138421507, 18.02708237518570082602587803133, 18.89713159934513791392403194344, 20.39776149114170323284831691553, 21.1458373101568420806207189832, 22.32609199519047196274368088946, 22.899554888225156954006037813539, 24.42278407025175321668245577103, 24.64927351987681090255702173641, 25.82669442632273157737275777420

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