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Properties

Label 1-253-253.190-r0-0-0
Degree $1$
Conductor $253$
Sign $0.465 + 0.885i$
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  + (−0.921 + 0.389i)2-s + (−0.998 + 0.0570i)3-s + (0.696 − 0.717i)4-s + (0.993 + 0.113i)5-s + (0.897 − 0.441i)6-s + (0.610 + 0.791i)7-s + (−0.362 + 0.931i)8-s + (0.993 − 0.113i)9-s + (−0.959 + 0.281i)10-s + (−0.654 + 0.755i)12-s + (−0.0285 + 0.999i)13-s + (−0.870 − 0.491i)14-s + (−0.998 − 0.0570i)15-s + (−0.0285 − 0.999i)16-s + (0.897 − 0.441i)17-s + (−0.870 + 0.491i)18-s + ⋯
L(s)  = 1  + (−0.921 + 0.389i)2-s + (−0.998 + 0.0570i)3-s + (0.696 − 0.717i)4-s + (0.993 + 0.113i)5-s + (0.897 − 0.441i)6-s + (0.610 + 0.791i)7-s + (−0.362 + 0.931i)8-s + (0.993 − 0.113i)9-s + (−0.959 + 0.281i)10-s + (−0.654 + 0.755i)12-s + (−0.0285 + 0.999i)13-s + (−0.870 − 0.491i)14-s + (−0.998 − 0.0570i)15-s + (−0.0285 − 0.999i)16-s + (0.897 − 0.441i)17-s + (−0.870 + 0.491i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 253 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(253\)    =    \(11 \cdot 23\)
Sign: $0.465 + 0.885i$
Analytic conductor: \(1.17492\)
Root analytic conductor: \(1.17492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{253} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 253,\ (0:\ ),\ 0.465 + 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6446530228 + 0.3893052970i\)
\(L(\frac12)\) \(\approx\) \(0.6446530228 + 0.3893052970i\)
\(L(1)\) \(\approx\) \(0.6668363427 + 0.2139898512i\)
\(L(1)\) \(\approx\) \(0.6668363427 + 0.2139898512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.921 + 0.389i)T \)
3 \( 1 + (-0.998 + 0.0570i)T \)
5 \( 1 + (0.993 + 0.113i)T \)
7 \( 1 + (0.610 + 0.791i)T \)
13 \( 1 + (-0.0285 + 0.999i)T \)
17 \( 1 + (0.897 - 0.441i)T \)
19 \( 1 + (-0.466 - 0.884i)T \)
29 \( 1 + (-0.466 + 0.884i)T \)
31 \( 1 + (-0.254 - 0.967i)T \)
37 \( 1 + (0.198 - 0.980i)T \)
41 \( 1 + (0.198 + 0.980i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.564 + 0.825i)T \)
59 \( 1 + (0.941 + 0.336i)T \)
61 \( 1 + (0.774 + 0.633i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (0.696 - 0.717i)T \)
79 \( 1 + (-0.0285 + 0.999i)T \)
83 \( 1 + (-0.736 + 0.676i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.736 - 0.676i)T \)
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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

−25.888849320685071630914942831367, −25.03070895052609995516501255490, −24.15495315280666053786746036049, −23.02550533664735156398727244974, −21.98950459240493024138515845935, −21.03423777546311862474394678573, −20.54153156682120736381008072157, −19.115004375342911086311827101740, −18.19671967417773950461834644185, −17.33553491077276937161769014955, −17.0536463544787581914572401000, −16.01246306589000329652400332701, −14.59928437229529254964409029902, −13.17955569704745210679942170148, −12.42159360112749746555502445115, −11.270330733740182381118097088714, −10.263456888338624957353469542647, −10.02603024361994338244477086462, −8.38298102023295623697068736860, −7.39792872062527102292212786065, −6.25667668556855500535856266608, −5.24692233816693254973166584524, −3.71442726266321528918106737505, −1.93410017212904779394127864515, −0.95192855803402492241991858267, 1.326606203733073318433203637191, 2.38234839020562137559217924267, 4.816004140850889986260318738376, 5.69623644166663986130008143005, 6.48740815209270357695896613807, 7.56133816836855943278822810223, 9.07465482815838510216180866459, 9.64359660417335034371247477250, 10.884782135621324436209674516069, 11.50156377310361712132281714571, 12.69158467687164392671037769675, 14.19850737770905733849406645428, 15.06004066926254699039952137241, 16.27558302369651746423092665918, 16.90336795109191931658756798956, 17.8966331183990370561546444492, 18.323931114482367636501017385463, 19.263870903970716383999968064828, 20.9168311102643377713805281144, 21.41191770857151429485018384312, 22.454193347897674995948559464562, 23.7633576871042418311774471370, 24.34075473845060576853601554514, 25.29349136161020818579784549433, 26.12920786245273366331468665210

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