Properties

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

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.841 + 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  :  $\chi_{92} (19, \cdot )$
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

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

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