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Properties

Label 1-253-253.153-r0-0-0
Degree $1$
Conductor $253$
Sign $0.294 + 0.955i$
Analytic cond. $1.17492$
Root an. cond. $1.17492$
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  + (0.959 + 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.142 − 0.989i)5-s + (−0.841 + 0.540i)6-s + (0.415 + 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−0.959 + 0.281i)12-s + (−0.415 + 0.909i)13-s + (0.142 + 0.989i)14-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.142 − 0.989i)5-s + (−0.841 + 0.540i)6-s + (0.415 + 0.909i)7-s + (0.654 + 0.755i)8-s + (−0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−0.959 + 0.281i)12-s + (−0.415 + 0.909i)13-s + (0.142 + 0.989i)14-s + (0.654 + 0.755i)15-s + (0.415 + 0.909i)16-s + (0.841 − 0.540i)17-s + (0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.294 + 0.955i$
Analytic conductor: \(1.17492\)
Root analytic conductor: \(1.17492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 253,\ (0:\ ),\ 0.294 + 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.518565434 + 1.120628394i\)
\(L(\frac12)\) \(\approx\) \(1.518565434 + 1.120628394i\)
\(L(1)\) \(\approx\) \(1.467413110 + 0.6354064129i\)
\(L(1)\) \(\approx\) \(1.467413110 + 0.6354064129i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.959 + 0.281i)T \)
3 \( 1 + (-0.654 + 0.755i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (0.841 + 0.540i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (0.142 + 0.989i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−25.49654662305834166872256094247, −24.647589440914537134947581873893, −23.671743856484341925956053332398, −23.086488140850560861724509792763, −22.32475335122147444981766446608, −21.53271133853169758057520371565, −20.240387250216910251134372954925, −19.46433785491248547631374160719, −18.42531073696500477697988862107, −17.50682075842064125659090878743, −16.51822866530151385716250182657, −15.19181922025483584336198474925, −14.27053308525521776643571954155, −13.53520568680899064543073990377, −12.591994709077051378374734966718, −11.536547143937781816936630816271, −10.76433541757193867143260898562, −10.08242965383493233408392805596, −7.61039204066175611186117596429, −7.22785704655945458470403187567, −6.00177761899474598818851178145, −5.17744509894607981064640504726, −3.72421989073810963797049590938, −2.53853954883177238440691759310, −1.20830485021642298930303671689, 1.76977074649753253836895256235, 3.41909327144299864150473518598, 4.6494651542419042042281027708, 5.29012581478281540528782650343, 6.05464972217885321236075290809, 7.55706604896969231257376224938, 8.88195383875025592205994531565, 9.82262727674900022444257658480, 11.42500143859673835346879508745, 11.92107197507140494561135078595, 12.740588788012729447877754691434, 14.11533933806990876138124925767, 14.94265084143991595478279636333, 16.00954391189289529885125902080, 16.535585641169397446682992374335, 17.39019964824289661783429913107, 18.69084970572887079280497810742, 20.37899592098850025298873923486, 20.83949917509579042098174765772, 21.74915557526499918351580268042, 22.3169692709294928079552245071, 23.51461239881694044229587839399, 24.18197368033674218360378232873, 24.99950575434004372932874087951, 25.99074758743120425264387832732

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