L-function 1-95-95.44-r0-0-0

• Label: 1-95-95.44-r0-0-0
• Degree: $1$
• Conductor: $95$
• Sign: $-0.486 + 0.873i$
• Analytic cond.: $0.441178$
• Root an. cond.: $0.441178$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.173 + 0.984i)2^-s + (−0.766 − 0.642i)3^-s + (−0.939 − 0.342i)4^-s + (0.766 − 0.642i)6^-s + (0.5 + 0.866i)7^-s + (0.5 − 0.866i)8^-s + (0.173 + 0.984i)9^-s + (−0.5 + 0.866i)11^-s + (0.5 + 0.866i)12^-s + (−0.766 + 0.642i)13^-s + (−0.939 + 0.342i)14^-s + (0.766 + 0.642i)16^-s + (−0.173 + 0.984i)17^-s − 18^-s + (0.173 − 0.984i)21^-s + (−0.766 − 0.642i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$95$    =    $5 \cdot 19$
Sign:	$-0.486 + 0.873i$
Analytic conductor:	$0.441178$
Root analytic conductor:	$0.441178$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{95} (44, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 95,\ (0:\ ),\ -0.486 + 0.873i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.2889849750 + 0.4918920032i$
$L(1)$	$\approx$	$0.5712607687 + 0.3520494375i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
good	2	$1 + (-0.173 + 0.984i)T$
	3	$1 + (-0.766 - 0.642i)T$
	7	$1 + (0.5 + 0.866i)T$
	11	$1 + (-0.5 + 0.866i)T$
	13	$1 + (-0.766 + 0.642i)T$
	17	$1 + (-0.173 + 0.984i)T$
	23	$1 + (0.939 + 0.342i)T$
	29	$1 + (0.173 + 0.984i)T$
	31	$1 + (-0.5 - 0.866i)T$

Imaginary part of the first few zeros on the critical line

−29.37030175171058875441614227092, −29.16730063606590436234174071161, −27.64810997870507466063725810964, −27.068313559407029138045022988210, −26.3367548921774362357736188656, −24.35724210473904135488017704306, −23.14029476890713997827917736935, −22.42397661831591506327381879477, −21.20122273275906246742395330275, −20.62072010004510285150972534033, −19.33186539661214580815328612633, −17.96892154417309495833182641252, −17.23513675972836087142021600612, −16.125059414909351747146022622641, −14.51171892997458936767333347085, −13.28397526172023901135217679034, −11.98580206652363561160016286628, −10.92522056111803779090221748709, −10.30037861370215781470781186142, −8.988717280386248030348546228166, −7.44581234608667745966945298625, −5.39509025107779972739445292015, −4.41209941781215047181457065867, −2.99082760385092158946060092064, −0.72177993422487286185860051996, 1.87740647536784338092634050845, 4.68125573538334042648971950858, 5.58016164358734039137170071963, 6.85672819430100289077290957776, 7.82920752897935063807843893933, 9.16244548123950335585367007225, 10.64955498492614543279382201327, 12.16042060369288096584766617984, 13.06161264587616843185824467831, 14.54133880145899064188023663080, 15.47915001017410849402160388316, 16.801151115103265659794186828, 17.63206016985073055640845262440, 18.48383722208035951503381822115, 19.43672250413615535431249409003, 21.46501134039593343283029555908, 22.40109401819842342588781886893, 23.509179197901480617429351607224, 24.25318694320516884053906855785, 25.12407242907385200242781438561, 26.1840959738156500326094856532, 27.60377891954124572948938275913, 28.24232568327549086491754093544, 29.27041479485375637103918637373, 30.93150659869267426805292676796

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