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Properties

Label 1-83-83.62-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.289 - 0.957i$
Analytic cond. $8.91958$
Root an. cond. $8.91958$
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  + (−0.988 + 0.152i)2-s + (0.0383 + 0.999i)3-s + (0.953 − 0.301i)4-s + (0.543 − 0.839i)5-s + (−0.190 − 0.981i)6-s + (0.338 − 0.941i)7-s + (−0.896 + 0.443i)8-s + (−0.997 + 0.0765i)9-s + (−0.409 + 0.912i)10-s + (−0.859 + 0.511i)11-s + (0.338 + 0.941i)12-s + (−0.720 − 0.693i)13-s + (−0.190 + 0.981i)14-s + (0.859 + 0.511i)15-s + (0.817 − 0.575i)16-s + (−0.665 − 0.746i)17-s + ⋯
L(s)  = 1  + (−0.988 + 0.152i)2-s + (0.0383 + 0.999i)3-s + (0.953 − 0.301i)4-s + (0.543 − 0.839i)5-s + (−0.190 − 0.981i)6-s + (0.338 − 0.941i)7-s + (−0.896 + 0.443i)8-s + (−0.997 + 0.0765i)9-s + (−0.409 + 0.912i)10-s + (−0.859 + 0.511i)11-s + (0.338 + 0.941i)12-s + (−0.720 − 0.693i)13-s + (−0.190 + 0.981i)14-s + (0.859 + 0.511i)15-s + (0.817 − 0.575i)16-s + (−0.665 − 0.746i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2963298638 - 0.3992936668i\)
\(L(\frac12)\) \(\approx\) \(0.2963298638 - 0.3992936668i\)
\(L(1)\) \(\approx\) \(0.6005951329 + 0.01906840444i\)
\(L(1)\) \(\approx\) \(0.6005951329 + 0.01906840444i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 \)
good2 \( 1 + (-0.988 + 0.152i)T \)
3 \( 1 + (0.0383 + 0.999i)T \)
5 \( 1 + (0.543 - 0.839i)T \)
7 \( 1 + (0.338 - 0.941i)T \)
11 \( 1 + (-0.859 + 0.511i)T \)
13 \( 1 + (-0.720 - 0.693i)T \)
17 \( 1 + (-0.665 - 0.746i)T \)
19 \( 1 + (-0.606 + 0.795i)T \)
23 \( 1 + (-0.973 - 0.227i)T \)
29 \( 1 + (-0.264 - 0.964i)T \)
31 \( 1 + (0.896 + 0.443i)T \)
37 \( 1 + (-0.997 - 0.0765i)T \)
41 \( 1 + (0.988 + 0.152i)T \)
43 \( 1 + (0.114 - 0.993i)T \)
47 \( 1 + (0.927 - 0.373i)T \)
53 \( 1 + (0.927 + 0.373i)T \)
59 \( 1 + (0.477 + 0.878i)T \)
61 \( 1 + (-0.771 + 0.636i)T \)
67 \( 1 + (-0.817 + 0.575i)T \)
71 \( 1 + (-0.338 - 0.941i)T \)
73 \( 1 + (0.409 - 0.912i)T \)
79 \( 1 + (-0.953 + 0.301i)T \)
89 \( 1 + (-0.190 - 0.981i)T \)
97 \( 1 + (-0.190 + 0.981i)T \)
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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

−30.62323343811869431574619462896, −29.66244716176562795792799223104, −28.9050629015615543608440947083, −27.96993903580645410841396323212, −26.30465311410571452289610065549, −25.90962482126499198810532149640, −24.6077737572615778745342918019, −23.94275439753301426656356388341, −22.05898292906852692376435231628, −21.21746976647009953247929697033, −19.57326685443858175850299277547, −18.82753977564813396449465738282, −18.00564073769382608091560634317, −17.24871002837850597798500286804, −15.55993221073172810657506505260, −14.37231696524994853023081706642, −12.90061036661722878403678798987, −11.66386038323031160838089545976, −10.69238669574890809294036369435, −9.14726229845444178792484762799, −8.06089441420925488661415432073, −6.826311586767495195228859829573, −5.82706989126744378032134395922, −2.698563325688577637932594518403, −1.96395415155262918546517215421, 0.29147927478525599675786373151, 2.32995633228678429182885794928, 4.4751673645628265353678869057, 5.7125624233218081906648550426, 7.59593520696339819459982656648, 8.698390532901194614693853081863, 10.028886683169165871999907740627, 10.445516167323175013076326264087, 12.08686845467654843409269490806, 13.81734471026634288000731515859, 15.20084662449295838443404251829, 16.214498346767335134198633349248, 17.16356959927906861639357749772, 17.85171051433339087897440362623, 19.75150541552290601936161988641, 20.55710618301214570587183821791, 21.11616626254989647535742431760, 22.8140429742640812292817384720, 24.17092954328024572292356316666, 25.232924082940915055428761332646, 26.28394370735399103261430060148, 27.10340437322880337552254989395, 28.03512570729000074375479291899, 28.91445795841275095052983443624, 29.90302261415690193955000860178

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