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Properties

Label 1-23-23.21-r1-0-0
Degree $1$
Conductor $23$
Sign $-0.997 + 0.0654i$
Analytic cond. $2.47169$
Root an. cond. $2.47169$
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  + (0.415 + 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.142 + 0.989i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (−0.415 + 0.909i)11-s + (0.415 − 0.909i)12-s + (−0.142 + 0.989i)13-s + (−0.841 + 0.540i)14-s + (0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (0.654 + 0.755i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.841 − 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.142 + 0.989i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (0.142 − 0.989i)10-s + (−0.415 + 0.909i)11-s + (0.415 − 0.909i)12-s + (−0.142 + 0.989i)13-s + (−0.841 + 0.540i)14-s + (0.959 + 0.281i)15-s + (−0.142 − 0.989i)16-s + (0.654 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(23\)
Sign: $-0.997 + 0.0654i$
Analytic conductor: \(2.47169\)
Root analytic conductor: \(2.47169\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{23} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 23,\ (1:\ ),\ -0.997 + 0.0654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02389792403 + 0.7289611314i\)
\(L(\frac12)\) \(\approx\) \(0.02389792403 + 0.7289611314i\)
\(L(1)\) \(\approx\) \(0.5095007080 + 0.5442675104i\)
\(L(1)\) \(\approx\) \(0.5095007080 + 0.5442675104i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (-0.959 + 0.281i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
11 \( 1 + (-0.415 + 0.909i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (0.654 + 0.755i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (-0.959 - 0.281i)T \)
37 \( 1 + (-0.841 + 0.540i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (-0.654 + 0.755i)T \)
79 \( 1 + (0.142 - 0.989i)T \)
83 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
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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

−38.504239193860861151722615080878, −36.94920847979221336197334833455, −35.57374246302482518030052360961, −34.246312678623398647916604288300, −32.87170351617143714049148322345, −31.32782148765533013536235653519, −29.94165050014673494663967477452, −29.39296573621252220254676150618, −27.64332499310706950267022878752, −26.921519015905945165008564526051, −24.1372744651950129518469277798, −23.147002474980256875151377821040, −22.321976908452752798497500746831, −20.59192156171817513312413194609, −19.12407163549371921750645334304, −18.00311216792705612564796344821, −16.13280970468500293150500055710, −14.16792048252041723320680372312, −12.61097450498563826340730970543, −11.23050428241421999713945822326, −10.39764921993340918893402839025, −7.523772029316227977289894112619, −5.42494992536204878958907862173, −3.556840269818296224727192548207, −0.595428854419990019860432101880, 4.31223141471866953244943236567, 5.56424305609979977796163425944, 7.38276781385186051167280960646, 9.20888671721732226819136023697, 11.73938477192518769181868185940, 12.66888442624885074371064169073, 15.04640302322557239389925993895, 15.95332477594523285053167929874, 17.19886726270929591888650616212, 18.64106230986563546797929956735, 21.06397724226405307327047145086, 22.31273365780189005852132945833, 23.54723092607088623459520491724, 24.3959365379030146891103417381, 26.15187241659205154632258317873, 27.65446962190666171919289977351, 28.49575722422457959364713864996, 30.70849605596497712260666767523, 31.759518285674679575627498354396, 33.06883409596838763447412914249, 34.31946760242251160255153711669, 35.065145920863293422096012634568, 36.26019274866078620162480300100, 38.641699042349634593485111211132, 39.47384591982704763404433687334

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