Properties

Label 1-283-283.128-r1-0-0
Degree $1$
Conductor $283$
Sign $0.486 - 0.873i$
Analytic cond. $30.4125$
Root an. cond. $30.4125$
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 + (0.944 − 0.328i)3-s + (0.860 + 0.509i)4-s + (−0.784 − 0.619i)5-s + (−0.997 + 0.0667i)6-s + (0.231 + 0.972i)7-s + (−0.695 − 0.718i)8-s + (0.784 − 0.619i)9-s + (0.593 + 0.805i)10-s + (0.860 − 0.509i)11-s + (0.979 + 0.199i)12-s + (−0.997 + 0.0667i)13-s + (0.0334 − 0.999i)14-s + (−0.944 − 0.328i)15-s + (0.480 + 0.876i)16-s + (0.0334 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.964 − 0.264i)2-s + (0.944 − 0.328i)3-s + (0.860 + 0.509i)4-s + (−0.784 − 0.619i)5-s + (−0.997 + 0.0667i)6-s + (0.231 + 0.972i)7-s + (−0.695 − 0.718i)8-s + (0.784 − 0.619i)9-s + (0.593 + 0.805i)10-s + (0.860 − 0.509i)11-s + (0.979 + 0.199i)12-s + (−0.997 + 0.0667i)13-s + (0.0334 − 0.999i)14-s + (−0.944 − 0.328i)15-s + (0.480 + 0.876i)16-s + (0.0334 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.486 - 0.873i$
Analytic conductor: \(30.4125\)
Root analytic conductor: \(30.4125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{283} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ 0.486 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.387627809 - 0.8157344867i\)
\(L(\frac12)\) \(\approx\) \(1.387627809 - 0.8157344867i\)
\(L(1)\) \(\approx\) \(0.9342330571 - 0.2708012619i\)
\(L(1)\) \(\approx\) \(0.9342330571 - 0.2708012619i\)

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 \( 1 + (0.944 - 0.328i)T \)
5 \( 1 + (-0.784 - 0.619i)T \)
7 \( 1 + (0.231 + 0.972i)T \)
11 \( 1 + (0.860 - 0.509i)T \)
13 \( 1 + (-0.997 + 0.0667i)T \)
17 \( 1 + (0.0334 + 0.999i)T \)
19 \( 1 + (0.997 - 0.0667i)T \)
23 \( 1 + (-0.420 - 0.907i)T \)
29 \( 1 + (0.920 + 0.390i)T \)
31 \( 1 + (-0.359 - 0.933i)T \)
37 \( 1 + (0.645 - 0.763i)T \)
41 \( 1 + (0.695 - 0.718i)T \)
43 \( 1 + (0.166 + 0.986i)T \)
47 \( 1 + (0.997 + 0.0667i)T \)
53 \( 1 + (-0.480 + 0.876i)T \)
59 \( 1 + (-0.892 + 0.451i)T \)
61 \( 1 + (0.593 - 0.805i)T \)
67 \( 1 + (0.979 - 0.199i)T \)
71 \( 1 + (0.860 - 0.509i)T \)
73 \( 1 + (0.359 - 0.933i)T \)
79 \( 1 + (0.166 - 0.986i)T \)
83 \( 1 + (-0.538 + 0.842i)T \)
89 \( 1 + (-0.166 - 0.986i)T \)
97 \( 1 + (-0.420 - 0.907i)T \)
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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

−25.65352026034580330911457586974, −24.872774665858767831965779844354, −23.97006940880081095489667669675, −22.9216202997308880657663902961, −21.78961221033634407165681886427, −20.376281320360961650128104886721, −19.962171044333831825896298388306, −19.34839382879299561528492980303, −18.2657543952844298016318629691, −17.30435085818888065972213948799, −16.196619496281050070178456639947, −15.472109768116054636261729409981, −14.452811659029534026856510275980, −14.00116520382958551088854095477, −12.100795474626777798862080063258, −11.19788869736185256931146157342, −10.02574925782143158049196066260, −9.54084958499519569126164184261, −8.17977406231337132474197373484, −7.367712853310221864399971833094, −6.92569954302342220769403255261, −4.86097370220016738991185096223, −3.6345134729804328572549410619, −2.52079934781418347708207536082, −1.02713254855383012016065973969, 0.756664366351916705011363576575, 1.98450023893817249885566624757, 3.096201081303930949507259901673, 4.272843957105723764689945085592, 6.101494733584078798989110500566, 7.422610551963939305753037473431, 8.178308126665511030825002745854, 8.9499462193779526578549326853, 9.60705126944814581772453293341, 11.14901205565266942218000210451, 12.27284208087061231481956560761, 12.52119301560930140667443517594, 14.28304600639159092058874167799, 15.166860615143180270955579969787, 16.010030323509308429282071204841, 17.00793191913606484494126943040, 18.14695865398321207287206797936, 19.035354951197366741953969453038, 19.63890656886030541776292026944, 20.27478560011917444240637281335, 21.33451323661574198793034882011, 22.13331969428346316675922670719, 24.01537707970516186202086473760, 24.55291696298360578327614887275, 25.08061891884508381715375712952

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