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Properties

Label 1-83-83.16-r0-0-0
Degree $1$
Conductor $83$
Sign $0.108 + 0.994i$
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  + (0.953 + 0.301i)2-s + (−0.997 − 0.0765i)3-s + (0.817 + 0.575i)4-s + (−0.409 + 0.912i)5-s + (−0.927 − 0.373i)6-s + (−0.771 + 0.636i)7-s + (0.606 + 0.795i)8-s + (0.988 + 0.152i)9-s + (−0.665 + 0.746i)10-s + (0.477 + 0.878i)11-s + (−0.771 − 0.636i)12-s + (0.0383 − 0.999i)13-s + (−0.927 + 0.373i)14-s + (0.477 − 0.878i)15-s + (0.338 + 0.941i)16-s + (−0.114 − 0.993i)17-s + ⋯
L(s)  = 1  + (0.953 + 0.301i)2-s + (−0.997 − 0.0765i)3-s + (0.817 + 0.575i)4-s + (−0.409 + 0.912i)5-s + (−0.927 − 0.373i)6-s + (−0.771 + 0.636i)7-s + (0.606 + 0.795i)8-s + (0.988 + 0.152i)9-s + (−0.665 + 0.746i)10-s + (0.477 + 0.878i)11-s + (−0.771 − 0.636i)12-s + (0.0383 − 0.999i)13-s + (−0.927 + 0.373i)14-s + (0.477 − 0.878i)15-s + (0.338 + 0.941i)16-s + (−0.114 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.108 + 0.994i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.108 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8399871680 + 0.7534274722i\)
\(L(\frac12)\) \(\approx\) \(0.8399871680 + 0.7534274722i\)
\(L(1)\) \(\approx\) \(1.084830588 + 0.5151792589i\)
\(L(1)\) \(\approx\) \(1.084830588 + 0.5151792589i\)

Euler product

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

−30.54109047042829928781114729161, −29.46617235095767548585095116585, −28.73505042521013929334814663758, −27.90241475886044119696883269466, −26.50813278749146272870503713402, −24.78767686975454684133021252068, −23.74598003168081179310172968739, −23.36033135600789325045973425138, −21.980459317378360395924569986545, −21.297946919346620363185182779911, −19.83763574280088623114078917024, −19.075491026768949882535782157273, −16.92992717412535092052918438328, −16.446925043832014013469996780008, −15.314795964494617396435046402591, −13.55135177588856720571012367277, −12.76215679274967254416311980877, −11.64288040406584098528050462216, −10.764002564676658695421133505382, −9.2365540845023234351159385853, −7.04809489648262711127535761227, −6.021380841807035724645473282596, −4.65285392671030571821777117750, −3.7010378639217365164770206012, −1.16681688387130208026942130459, 2.62640004241850237489589569044, 4.12232556093047870072922901928, 5.63150432903936505928451646650, 6.5961241346014327100004416570, 7.5854179503104658616190213661, 9.95932141382335657338106187774, 11.249776845364752603156412824139, 12.1866415841278246846260718637, 13.09369992721623849392782097956, 14.81027750558449773889006111797, 15.543415209851775233070472891014, 16.641698592511709693427372447970, 17.93535160523014940391987296786, 19.093351392803320613287914476767, 20.59799960566801378957947800194, 22.03444950504501380259370922697, 22.830715038103940050709430570971, 22.9867022990939111779064076869, 24.69155692362513986116800928237, 25.44939406149688683167704264108, 26.87771750910400876346394960440, 28.10019777699950841651682539325, 29.34681339909836591582464624553, 30.05338894261507242592175564300, 31.098083312925532143422918435284

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