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Properties

Label 1-83-83.43-r1-0-0
Degree $1$
Conductor $83$
Sign $-0.675 - 0.737i$
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  + (0.665 − 0.746i)2-s + (−0.543 + 0.839i)3-s + (−0.114 − 0.993i)4-s + (−0.720 + 0.693i)5-s + (0.264 + 0.964i)6-s + (0.896 − 0.443i)7-s + (−0.817 − 0.575i)8-s + (−0.409 − 0.912i)9-s + (0.0383 + 0.999i)10-s + (0.190 − 0.981i)11-s + (0.896 + 0.443i)12-s + (−0.477 − 0.878i)13-s + (0.264 − 0.964i)14-s + (−0.190 − 0.981i)15-s + (−0.973 + 0.227i)16-s + (−0.997 + 0.0765i)17-s + ⋯
L(s)  = 1  + (0.665 − 0.746i)2-s + (−0.543 + 0.839i)3-s + (−0.114 − 0.993i)4-s + (−0.720 + 0.693i)5-s + (0.264 + 0.964i)6-s + (0.896 − 0.443i)7-s + (−0.817 − 0.575i)8-s + (−0.409 − 0.912i)9-s + (0.0383 + 0.999i)10-s + (0.190 − 0.981i)11-s + (0.896 + 0.443i)12-s + (−0.477 − 0.878i)13-s + (0.264 − 0.964i)14-s + (−0.190 − 0.981i)15-s + (−0.973 + 0.227i)16-s + (−0.997 + 0.0765i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.675 - 0.737i$
Analytic conductor: \(8.91958\)
Root analytic conductor: \(8.91958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (1:\ ),\ -0.675 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4742898903 - 1.077873371i\)
\(L(\frac12)\) \(\approx\) \(0.4742898903 - 1.077873371i\)
\(L(1)\) \(\approx\) \(0.9225547616 - 0.4422790722i\)
\(L(1)\) \(\approx\) \(0.9225547616 - 0.4422790722i\)

Euler product

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

−31.09605903845581068537531506160, −30.36747413705287147444104447564, −28.91046283257540559154633180553, −27.8584278613703143247049187508, −26.73559590987494191018677004043, −25.0117887184132053382102046366, −24.59123341778228398626340158517, −23.57624282696164269079615137879, −22.870996539672918884611993372226, −21.546642335971833138368998065, −20.321947366217439037268725988277, −18.86725374201512032863753802421, −17.54244439062111879128553254341, −16.86452047261511748115275280978, −15.52673809839532028662057323670, −14.451788117767660134379904618881, −13.07081311318845134422742863641, −12.11096036018282596106345720046, −11.45256944885311779006723661804, −8.84318023891409901502624599019, −7.76486389222526564732806706538, −6.784172864134727800227658917459, −5.22409733676094735016528062060, −4.36727810183604151622735899642, −1.95393013021167537673326980814, 0.47662007499814702052013852015, 2.94539367723187023724193950786, 4.15555165931202445264518946500, 5.2071413767296651576237022062, 6.75041920378643476648516021554, 8.72309391291385331569622192401, 10.47598915970516342433006339695, 11.02613963804029449372824121290, 11.86185509766517424116118551583, 13.53805573897517718394492990967, 14.90333093208374548241533135611, 15.39320024252479690500003319994, 17.08667110579407341418718429821, 18.34902803545136910600800354630, 19.69217234446161883476917288477, 20.61451815904537720066684022734, 21.82004709539735560806935962905, 22.43742971242981973294122425408, 23.53434618437283150681624044131, 24.3544347178869280507164914407, 26.56535438413316052178264830766, 27.22295605144344050730253063362, 28.01158620441852712123910536222, 29.3689029546201783235222480860, 30.18430192680673666476861767156

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