Properties

Degree 1
Conductor 97
Sign $0.0711 - 0.997i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.793 − 0.608i)2-s + (0.991 + 0.130i)3-s + (0.258 − 0.965i)4-s + (−0.751 − 0.659i)5-s + (0.866 − 0.5i)6-s + (0.946 + 0.321i)7-s + (−0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (−0.997 − 0.0654i)10-s + (0.608 − 0.793i)11-s + (0.382 − 0.923i)12-s + (−0.659 + 0.751i)13-s + (0.946 − 0.321i)14-s + (−0.659 − 0.751i)15-s + (−0.866 − 0.5i)16-s + (−0.321 − 0.946i)17-s + ⋯
L(s,χ)  = 1  + (0.793 − 0.608i)2-s + (0.991 + 0.130i)3-s + (0.258 − 0.965i)4-s + (−0.751 − 0.659i)5-s + (0.866 − 0.5i)6-s + (0.946 + 0.321i)7-s + (−0.382 − 0.923i)8-s + (0.965 + 0.258i)9-s + (−0.997 − 0.0654i)10-s + (0.608 − 0.793i)11-s + (0.382 − 0.923i)12-s + (−0.659 + 0.751i)13-s + (0.946 − 0.321i)14-s + (−0.659 − 0.751i)15-s + (−0.866 − 0.5i)16-s + (−0.321 − 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.0711 - 0.997i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.0711 - 0.997i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $0.0711 - 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (26, \cdot )$
Sato-Tate  :  $\mu(96)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (1:\ ),\ 0.0711 - 0.997i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.566377588 - 2.389804748i$
$L(\frac12,\chi)$  $\approx$  $2.566377588 - 2.389804748i$
$L(\chi,1)$  $\approx$  1.910776600 - 1.032425283i
$L(1,\chi)$  $\approx$  1.910776600 - 1.032425283i

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

−30.65488685718328289924048366148, −29.70872753674888973451405654442, −27.48500615904997898110398250727, −26.801457250140287219844255069368, −25.80076723638343334010511896091, −24.7624122117959775042844061484, −23.97008855248296281879004455796, −22.858452404326433515161042399274, −21.82695555525794537421086864821, −20.51989105788541249300822887058, −19.85157376892391103099553703394, −18.33097273972279175613960688146, −17.18386153294650387009592571964, −15.61249914245203671147752384301, −14.66703479643336263704629721463, −14.39302006513342960675610756268, −12.81791572621635047876260892647, −11.8209561542628999583622152159, −10.262296889249747401376221738, −8.287907921680189117081697609088, −7.711393575848332159575496806768, −6.58930388684537665845805908099, −4.57070528304514631358299543447, −3.6813945527209434745507590745, −2.20201023800986516795992536586, 1.28241226385783953275788096861, 2.792872056934194360186715710630, 4.190418725850727422444768882016, 5.028313485623048409641205773709, 7.107664783971295561400270444557, 8.568068032772781256454196164946, 9.50648099402571333890149571917, 11.30379255980877432432419644139, 11.99072212744217144008061397045, 13.44451496819156849577055704101, 14.28713298404360560757866576983, 15.28555674096919913723741550776, 16.28071132827626755176530509934, 18.273930226758349578883473281362, 19.64820653131238338799007536066, 19.89784532280111705658308486347, 21.297645472117740126977647979508, 21.75431056525706574590315659411, 23.49279988221505508200225647354, 24.40304189187732333762072709001, 24.89516885607498277613864617290, 26.86315484247231050995654570822, 27.444539004459006899652749251, 28.61860864015568347989951951716, 29.94478091343079207256961130534

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