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Properties

Label 1-83-83.65-r0-0-0
Degree $1$
Conductor $83$
Sign $-0.992 - 0.122i$
Analytic cond. $0.385450$
Root an. cond. $0.385450$
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  + (−0.114 + 0.993i)2-s + (−0.409 + 0.912i)3-s + (−0.973 − 0.227i)4-s + (0.0383 + 0.999i)5-s + (−0.859 − 0.511i)6-s + (0.606 + 0.795i)7-s + (0.338 − 0.941i)8-s + (−0.665 − 0.746i)9-s + (−0.997 − 0.0765i)10-s + (−0.927 + 0.373i)11-s + (0.606 − 0.795i)12-s + (−0.543 − 0.839i)13-s + (−0.859 + 0.511i)14-s + (−0.927 − 0.373i)15-s + (0.896 + 0.443i)16-s + (0.988 + 0.152i)17-s + ⋯
L(s)  = 1  + (−0.114 + 0.993i)2-s + (−0.409 + 0.912i)3-s + (−0.973 − 0.227i)4-s + (0.0383 + 0.999i)5-s + (−0.859 − 0.511i)6-s + (0.606 + 0.795i)7-s + (0.338 − 0.941i)8-s + (−0.665 − 0.746i)9-s + (−0.997 − 0.0765i)10-s + (−0.927 + 0.373i)11-s + (0.606 − 0.795i)12-s + (−0.543 − 0.839i)13-s + (−0.859 + 0.511i)14-s + (−0.927 − 0.373i)15-s + (0.896 + 0.443i)16-s + (0.988 + 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.992 - 0.122i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ -0.992 - 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04060229313 + 0.6624484209i\)
\(L(\frac12)\) \(\approx\) \(-0.04060229313 + 0.6624484209i\)
\(L(1)\) \(\approx\) \(0.4001113457 + 0.6499987064i\)
\(L(1)\) \(\approx\) \(0.4001113457 + 0.6499987064i\)

Euler product

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

−30.04919434514879042117705442842, −29.29397067763079290747084569560, −28.48897567521654177820489650755, −27.503228960693252622803371104135, −26.31560423580999385628991524276, −24.72741771622603398233630018382, −23.67676870597943541379740637862, −23.0634544550637557631574633286, −21.31472341321882462354477868800, −20.66421924001962391090255242102, −19.390294429793335075587093061185, −18.56838601565881332122940726541, −17.203245833687928541061178216761, −16.71330687931260520320363632305, −14.22689820500002673861046685878, −13.28631874699172633246500163185, −12.36974428561813463654894693633, −11.37173983308872629857222380799, −10.154259753862826735224589576702, −8.51815685059808962039461215924, −7.60604275630453727246161320644, −5.49504274862627657025655298053, −4.34196543319661365385298450863, −2.209082674132610485504531953910, −0.83858215163641765247996372360, 3.0706717827445151023425083166, 4.9088260677666174937202027337, 5.70585947245633419473350689272, 7.214727303042291056681174688389, 8.55237990746564907354988156564, 9.957579733012927911041715114347, 10.83090162681449778086498143391, 12.500106779571776146369959449907, 14.32152780947631765549383387483, 15.14137698623849994620938759624, 15.717284975309664491189727678964, 17.32583641328052452039311750621, 17.94934388662329961925826880688, 19.14179741370469088696945910474, 21.094296235905738891378486671031, 21.96131177406741701658653036071, 22.90567785650568212648761750200, 23.79291373166627490757285729638, 25.39981982673803845046532075093, 26.00089712589748675469231729820, 27.2965611140673209996232030296, 27.69259295690442427556444241915, 29.0689063423034510119138395234, 30.69713617612388662444984140590, 31.68275159117371509209388372664

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