Properties

Degree 1
Conductor 103
Sign $-0.215 + 0.976i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.650 + 0.759i)2-s + (0.0922 + 0.995i)3-s + (−0.153 + 0.988i)4-s + (0.992 − 0.122i)5-s + (−0.696 + 0.717i)6-s + (0.881 − 0.473i)7-s + (−0.850 + 0.526i)8-s + (−0.982 + 0.183i)9-s + (0.739 + 0.673i)10-s + (0.332 − 0.943i)11-s + (−0.998 − 0.0615i)12-s + (−0.850 − 0.526i)13-s + (0.932 + 0.361i)14-s + (0.213 + 0.976i)15-s + (−0.952 − 0.303i)16-s + (−0.696 − 0.717i)17-s + ⋯
L(s,χ)  = 1  + (0.650 + 0.759i)2-s + (0.0922 + 0.995i)3-s + (−0.153 + 0.988i)4-s + (0.992 − 0.122i)5-s + (−0.696 + 0.717i)6-s + (0.881 − 0.473i)7-s + (−0.850 + 0.526i)8-s + (−0.982 + 0.183i)9-s + (0.739 + 0.673i)10-s + (0.332 − 0.943i)11-s + (−0.998 − 0.0615i)12-s + (−0.850 − 0.526i)13-s + (0.932 + 0.361i)14-s + (0.213 + 0.976i)15-s + (−0.952 − 0.303i)16-s + (−0.696 − 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.215 + 0.976i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.215 + 0.976i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(103\)
\( \varepsilon \)  =  $-0.215 + 0.976i$
motivic weight  =  \(0\)
character  :  $\chi_{103} (33, \cdot )$
Sato-Tate  :  $\mu(51)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 103,\ (0:\ ),\ -0.215 + 0.976i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.9929917080 + 1.235558488i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.9929917080 + 1.235558488i\)
\(L(\chi,1)\)  \(\approx\)  \(1.225005191 + 0.9465037722i\)
\(L(1,\chi)\)  \(\approx\)  \(1.225005191 + 0.9465037722i\)

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

−29.83080645757086298964031624334, −28.60949110737421116581254047280, −28.082132141523199257362123339948, −26.29589780063594053600233564181, −24.93927251641818915093459188143, −24.38343888189157712656078053365, −23.2927542046733562938388562136, −22.05741504061528238577888640560, −21.28923855004737826134120685649, −20.083252190444575840030858428833, −19.14851851592429417636119329591, −17.91689088913741395182099478212, −17.40316038412069414248245633174, −14.95143645126923321156176866227, −14.34426303870097565178142790564, −13.231619051729269295640358805493, −12.30996976055218791731471306305, −11.33412841613099679032642348, −9.9172053339943244263570523214, −8.68002939198357995533169695269, −6.87505755355599233480130359873, −5.80684936973811115927279086021, −4.49939276416839696566749872165, −2.25650891670276969814875890957, −1.88233119473528458999127937938, 2.64014859037645278115370700799, 4.2371847439787004229940648831, 5.187564228559758072197923216263, 6.27199181889312935905792793631, 8.00744403971961772272720324061, 9.06410946935062949225129235102, 10.42021714697035230021541679327, 11.6768584028346049665728517769, 13.34211192902799462932789367875, 14.244871960662246641635997393923, 14.9391964484153066254193835172, 16.3788723081014284740432244592, 17.061609689639070202757636383371, 17.97943637167788395711577040533, 20.13603833555008079521006541353, 21.05963862027546685093558482237, 21.83388022605288684570935439226, 22.592814117292056175407272489967, 24.09285446983861734670003099174, 24.85175172917922619377124166340, 25.95821279377054044955851796121, 26.87596650084128813785495652635, 27.64315917401023473511641033927, 29.343247560014340488621355119544, 30.12027477347905117949764674933

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