Properties

Degree 1
Conductor 97
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯
L(s,χ)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(97\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{97} (96, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 97,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.943996192$
$L(\frac12,\chi)$  $\approx$  $1.943996192$
$L(\chi,1)$  $\approx$  1.893495323
$L(1,\chi)$  $\approx$  1.893495323

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

−30.26966459451872039316313201513, −29.55975815890733306187645621718, −28.05239012591340682006811668163, −26.7423449736172101876969842674, −25.80611663339547860278899419772, −24.69360431769096635791079466598, −23.96179064651779391335593418256, −22.59219451092514543203786014080, −21.93422738587998203728503530778, −20.43187054233479097209448063531, −19.61956313941108531457018242411, −19.16307561130216494674426449551, −16.842355872364980851935272471445, −15.66064621525898780938462147556, −15.00704193792237262796158440501, −13.89874205955462555010939888256, −12.73750976794474924787776389717, −11.910298558178122093760965782721, −10.31886818510980438157940710570, −8.86113553943907597146249210103, −7.41458435128604388582444775025, −6.51632663856962226607151954769, −4.39405248323803031496271391218, −3.62811412250984882597424611474, −2.31457864518104887525881513152, 2.31457864518104887525881513152, 3.62811412250984882597424611474, 4.39405248323803031496271391218, 6.51632663856962226607151954769, 7.41458435128604388582444775025, 8.86113553943907597146249210103, 10.31886818510980438157940710570, 11.910298558178122093760965782721, 12.73750976794474924787776389717, 13.89874205955462555010939888256, 15.00704193792237262796158440501, 15.66064621525898780938462147556, 16.842355872364980851935272471445, 19.16307561130216494674426449551, 19.61956313941108531457018242411, 20.43187054233479097209448063531, 21.93422738587998203728503530778, 22.59219451092514543203786014080, 23.96179064651779391335593418256, 24.69360431769096635791079466598, 25.80611663339547860278899419772, 26.7423449736172101876969842674, 28.05239012591340682006811668163, 29.55975815890733306187645621718, 30.26966459451872039316313201513

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