L-function 1-319-319.128-r0-0-0

• Label: 1-319-319.128-r0-0-0
• Degree: $1$
• Conductor: $319$
• Sign: $0.421 - 0.906i$
• Analytic cond.: $1.48142$
• Root an. cond.: $1.48142$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.951 − 0.309i)2^-s + (0.587 − 0.809i)3^-s + (0.809 − 0.587i)4^-s + (−0.309 + 0.951i)5^-s + (0.309 − 0.951i)6^-s + (0.809 − 0.587i)7^-s + (0.587 − 0.809i)8^-s + (−0.309 − 0.951i)9^-s + i·10^-s − i·12^-s + (0.309 + 0.951i)13^-s + (0.587 − 0.809i)14^-s + (0.587 + 0.809i)15^-s + (0.309 − 0.951i)16^-s + (−0.951 − 0.309i)17^-s + (−0.587 − 0.809i)18^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$319$    =    $11 \cdot 29$
Sign:	$0.421 - 0.906i$
Analytic conductor:	$1.48142$
Root analytic conductor:	$1.48142$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{319} (128, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 319,\ (0:\ ),\ 0.421 - 0.906i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.298431607 - 1.466611677i$
$L(1)$	$\approx$	$1.969713667 - 0.8158712504i$

Euler product

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

	$p$	$F_p(T)$
bad	11	$1$
	29	$1$
good	2	$1 + (0.951 - 0.309i)T$
	3	$1 + (0.587 - 0.809i)T$
	5	$1 + (-0.309 + 0.951i)T$
	7	$1 + (0.809 - 0.587i)T$
	13	$1 + (0.309 + 0.951i)T$
	17	$1 + (-0.951 - 0.309i)T$
	19	$1 + (-0.587 + 0.809i)T$
	23	$1 + T$
	31	$1 + (-0.951 + 0.309i)T$

Imaginary part of the first few zeros on the critical line

−25.00237898556459419998129966055, −24.58901508829854979369877002502, −23.58426963450734624364356767399, −22.57747145138544992440696822034, −21.520687990560396870905131242158, −21.1226602864384075809816680625, −20.15191338529568606403036078333, −19.62615022030843978781150000903, −17.834685673446630973194356023147, −16.861148676672509600677495695405, −15.94097377330563390137560607762, −15.17636481883180801406917092836, −14.71198631380233332620819729679, −13.30107900393986176794478476096, −12.8340265758598796186755955286, −11.429788928905716553275286191677, −10.8381435332592640567699379587, −9.11667428577254030258629567808, −8.425506515595833227528960657152, −7.58473074442533043872991905223, −5.88228384298036932340527115346, −4.915529016063408451616918498513, −4.3681585376316529177396228753, −3.11833956380532663456438694602, −1.94281547192344600712974598763, 1.49334617639516090936940977479, 2.43878514299481468885099256884, 3.596803797426690241409077111186, 4.48350955971953564978303327662, 6.12502237622174983141798980601, 6.96805168158769854964504458124, 7.645497047813495114145122233518, 9.02690377279753231734951527197, 10.58178451733003498160650592083, 11.27066081075708179731598424245, 12.125862466198467121416519670789, 13.30512625633963677192836579118, 14.03486857014266607311350589965, 14.642604688912718148787987242, 15.4208154398111197013500025217, 16.83720585381644720730244905728, 18.13720619515626823664499352519, 18.89248090630589851780717906545, 19.68032552484494361339064230370, 20.58701630701660332454490232023, 21.32919346542754795251309057233, 22.46173104709271562072890532479, 23.429493994820739337646284176551, 23.81497634419343336069901809679, 24.79525161367746171471168023551

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