Properties

Degree 1
Conductor 83
Sign $0.129 - 0.991i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.129 - 0.991i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (25, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ 0.129 - 0.991i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4156126311 - 0.3647075428i$
$L(\frac12,\chi)$  $\approx$  $0.4156126311 - 0.3647075428i$
$L(\chi,1)$  $\approx$  0.5797910833 - 0.2599873937i
$L(1,\chi)$  $\approx$  0.5797910833 - 0.2599873937i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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