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Properties

Label 1-83-83.25-r0-0-0
Degree $1$
Conductor $83$
Sign $0.129 - 0.991i$
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  + (−0.543 − 0.839i)2-s + (−0.859 + 0.511i)3-s + (−0.409 + 0.912i)4-s + (0.190 − 0.981i)5-s + (0.896 + 0.443i)6-s + (−0.114 + 0.993i)7-s + (0.988 − 0.152i)8-s + (0.477 − 0.878i)9-s + (−0.927 + 0.373i)10-s + (0.338 − 0.941i)11-s + (−0.114 − 0.993i)12-s + (−0.264 − 0.964i)13-s + (0.896 − 0.443i)14-s + (0.338 + 0.941i)15-s + (−0.665 − 0.746i)16-s + (0.720 − 0.693i)17-s + ⋯
L(s)  = 1  + (−0.543 − 0.839i)2-s + (−0.859 + 0.511i)3-s + (−0.409 + 0.912i)4-s + (0.190 − 0.981i)5-s + (0.896 + 0.443i)6-s + (−0.114 + 0.993i)7-s + (0.988 − 0.152i)8-s + (0.477 − 0.878i)9-s + (−0.927 + 0.373i)10-s + (0.338 − 0.941i)11-s + (−0.114 − 0.993i)12-s + (−0.264 − 0.964i)13-s + (0.896 − 0.443i)14-s + (0.338 + 0.941i)15-s + (−0.665 − 0.746i)16-s + (0.720 − 0.693i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.129 - 0.991i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.129 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4156126311 - 0.3647075428i\)
\(L(\frac12)\) \(\approx\) \(0.4156126311 - 0.3647075428i\)
\(L(1)\) \(\approx\) \(0.5797910833 - 0.2599873937i\)
\(L(1)\) \(\approx\) \(0.5797910833 - 0.2599873937i\)

Euler product

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

−30.91902680942884323383833130371, −29.91259701316358693091573200026, −28.91949380030799963590935731925, −27.8747427546075293383735903311, −26.713353143697610539949179772, −25.917088907782028368736479260573, −24.72990170263943558759965653829, −23.50210255769858428955576538554, −22.99735773760700752318567653035, −21.89045996127224130719110695513, −19.83330884070941165791424022884, −18.83001538794152303087685986483, −17.827173846587573868532936102142, −17.08402597643467518345099264938, −16.03483117747618418854031554631, −14.4928751352209610838978121684, −13.70642995869781184929384686529, −11.95372695347614230433813277925, −10.5254207984264509562090915442, −9.82928240211027948209805246694, −7.64832098484825997984053915299, −6.96731998522032542566989707440, −5.988178714626572361349500775423, −4.34682089578950979619990160796, −1.59445346606149419454705161491, 0.91824683336120472237200705249, 3.0741958113842928298261713068, 4.76948870455811270162925944102, 5.87121387435356280016009307147, 8.12330268861424852044986533119, 9.2888945003443039075910340183, 10.16260299994409102072504878865, 11.78283137181396263340378637065, 12.14109666953916387081598667673, 13.570513498642519249403896520, 15.70360674506094899923742117673, 16.53015569956707242496069527446, 17.59518695045553348101936768883, 18.52761245505559061247807165398, 19.93498985053718457795821000221, 21.01535769557017894977171780307, 21.83247458692516097585055500417, 22.68428481801098344958954642860, 24.31753045099828928945203149947, 25.36364077234717449790943565613, 26.97110369621244151940380161180, 27.64104496891165649034863971461, 28.55058358707813898093806573002, 29.16208967705065921614352080388, 30.30494805835673317532144386941

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