Properties

Degree 1
Conductor $ 2^{2} \cdot 23 $
Sign $-0.529 + 0.848i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.841 − 0.540i)3-s + (−0.415 + 0.909i)5-s + (−0.959 − 0.281i)7-s + (0.415 + 0.909i)9-s + (−0.654 + 0.755i)11-s + (−0.959 + 0.281i)13-s + (0.841 − 0.540i)15-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.959 − 0.281i)33-s + ⋯
L(s,χ)  = 1  + (−0.841 − 0.540i)3-s + (−0.415 + 0.909i)5-s + (−0.959 − 0.281i)7-s + (0.415 + 0.909i)9-s + (−0.654 + 0.755i)11-s + (−0.959 + 0.281i)13-s + (0.841 − 0.540i)15-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (0.654 + 0.755i)21-s + (−0.654 − 0.755i)25-s + (0.142 − 0.989i)27-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.959 − 0.281i)33-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.529 + 0.848i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.529 + 0.848i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(92\)    =    \(2^{2} \cdot 23\)
\( \varepsilon \)  =  $-0.529 + 0.848i$
motivic weight  =  \(0\)
character  :  $\chi_{92} (63, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 92,\ (0:\ ),\ -0.529 + 0.848i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1551232152 + 0.2796552153i$
$L(\frac12,\chi)$  $\approx$  $0.1551232152 + 0.2796552153i$
$L(\chi,1)$  $\approx$  0.5229382686 + 0.1117682732i
$L(1,\chi)$  $\approx$  0.5229382686 + 0.1117682732i

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

−29.45909134661663948990517400407, −29.030179980300020457040968289387, −27.9216682487458470966516850560, −27.121336220515999843011996276562, −25.96661243964300095779115892389, −24.50166195637631133685677385161, −23.6709687316976729838201211791, −22.53872928329395822739797025709, −21.681079748101642426590701293, −20.50052509078913426200703268387, −19.387490338684116456614952059867, −18.07767512065175902531650771759, −16.717262238629499657658784716934, −16.14199211319023275741031958075, −15.189807976408408074684756468859, −13.23457496536436956667231097241, −12.3336182030780045619428912345, −11.25975850396344579921522960015, −9.873743632224684809438940383494, −8.90663128144168683424554825196, −7.191955402242423269207627402780, −5.65114230570768318193697414329, −4.74827270958926786856033451582, −3.140183411876865885892168469970, −0.352483842504547713766268536840, 2.27254551616905060094306387621, 4.014667630872880653147363828494, 5.75789386477998485789795657968, 6.921291383407553460109583067336, 7.71329759137789689917021661464, 9.947520574994147788937940137290, 10.706789616896632805817672476343, 12.13458686829973182305261396037, 12.8886398402026561483030381426, 14.39483766203505640027147040956, 15.6626981599379492174580007191, 16.79268857223192537694977875496, 17.87929940277061844942469434356, 18.980481155368042471425195010674, 19.63835997610317662325721307639, 21.45857314478167431092060445785, 22.66879140891164215716449075569, 23.060426993132512604943718153320, 24.188891148461168386566529435435, 25.58077791390157667919220160382, 26.48552409171123157796280036461, 27.67781028664894968196360173756, 28.85571041400772221673003517471, 29.55516131077395107690710887848, 30.558758168291631942675282070345

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