Properties

Degree 1
Conductor 97
Sign $-0.459 + 0.888i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.991 + 0.130i)2-s + (0.793 + 0.608i)3-s + (0.965 − 0.258i)4-s + (−0.896 − 0.442i)5-s + (−0.866 − 0.5i)6-s + (0.0654 − 0.997i)7-s + (−0.923 + 0.382i)8-s + (0.258 + 0.965i)9-s + (0.946 + 0.321i)10-s + (−0.130 + 0.991i)11-s + (0.923 + 0.382i)12-s + (−0.442 + 0.896i)13-s + (0.0654 + 0.997i)14-s + (−0.442 − 0.896i)15-s + (0.866 − 0.5i)16-s + (0.997 − 0.0654i)17-s + ⋯
L(s,χ)  = 1  + (−0.991 + 0.130i)2-s + (0.793 + 0.608i)3-s + (0.965 − 0.258i)4-s + (−0.896 − 0.442i)5-s + (−0.866 − 0.5i)6-s + (0.0654 − 0.997i)7-s + (−0.923 + 0.382i)8-s + (0.258 + 0.965i)9-s + (0.946 + 0.321i)10-s + (−0.130 + 0.991i)11-s + (0.923 + 0.382i)12-s + (−0.442 + 0.896i)13-s + (0.0654 + 0.997i)14-s + (−0.442 − 0.896i)15-s + (0.866 − 0.5i)16-s + (0.997 − 0.0654i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.459 + 0.888i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.459 + 0.888i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $-0.459 + 0.888i$
motivic weight  =  \(0\)
character  :  $\chi_{97} (57, \cdot )$
Sato-Tate  :  $\mu(96)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (1:\ ),\ -0.459 + 0.888i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4477187226 + 0.7359265030i$
$L(\frac12,\chi)$  $\approx$  $0.4477187226 + 0.7359265030i$
$L(\chi,1)$  $\approx$  0.6898184951 + 0.2586465909i
$L(1,\chi)$  $\approx$  0.6898184951 + 0.2586465909i

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

−29.712735901455658318832071941896, −28.28624807220856669134363704457, −27.29347395964114249229083274385, −26.47461517777750529740441577333, −25.40823683957672614408029053505, −24.61516075411811822339236750936, −23.598364383339373641508499424228, −21.88980568856898216400009388697, −20.70558443660403675657809542959, −19.59129142199283846069912769848, −18.841223855870429499528326427268, −18.30871708177448618774205128888, −16.73966513901711523057862015727, −15.30775141940637155514688400937, −14.82712980276241254022903191350, −12.856721978913677830595794381419, −11.89706730728402458710266710582, −10.735977674749915285979968792091, −9.15584059957926479443538961524, −8.23495142541214851069517379008, −7.443147712578242248945348189485, −6.0285830201773741745392718154, −3.35086803085185886905623050902, −2.44503484594241922902741717513, −0.48147420671365513060644592707, 1.61501648512957763488704388499, 3.49958683067354996898912971989, 4.785301105270811693264132643422, 7.18482934291289894255120846007, 7.82523592030446442086986038464, 9.09899881181905421607572953082, 10.05916278197457491501537316771, 11.17501364456038024980017617687, 12.60381898172051454298626929795, 14.40422438058613271191460761908, 15.228494551932313259164154680061, 16.443817108991170949274320951312, 17.01199285114731061900600380614, 18.80626022912423724849927347315, 19.64091834864143813080580235076, 20.41997212415486328522324624200, 21.19911041567974527014512534121, 23.15251359734032971348850479949, 24.05981229659638316297086651720, 25.384970462292823434987364083647, 26.121775057635126688692457571944, 27.2705001845752173126506070801, 27.55576496323306629633304326088, 28.88495695233967002075626507618, 30.1897346003795122076723842821

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