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Properties

Label 1-92-92.63-r0-0-0
Degree $1$
Conductor $92$
Sign $-0.529 + 0.848i$
Analytic cond. $0.427246$
Root an. cond. $0.427246$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(92\)    =    \(2^{2} \cdot 23\)
Sign: $-0.529 + 0.848i$
Analytic conductor: \(0.427246\)
Root analytic conductor: \(0.427246\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{92} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 92,\ (0:\ ),\ -0.529 + 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1551232152 + 0.2796552153i\)
\(L(\frac12)\) \(\approx\) \(0.1551232152 + 0.2796552153i\)
\(L(1)\) \(\approx\) \(0.5229382686 + 0.1117682732i\)
\(L(1)\) \(\approx\) \(0.5229382686 + 0.1117682732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (-0.959 - 0.281i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (-0.415 - 0.909i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 - T \)
53 \( 1 + (0.959 + 0.281i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (0.415 + 0.909i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.415 + 0.909i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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