L-function 1-253-253.190-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	11	$1$
	23	$1$
good	2	$1 + (-0.921 + 0.389i)T$
	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	$1 + (-0.254 - 0.967i)T$

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