L-function 1-319-319.173-r1-0-0

• Label: 1-319-319.173-r1-0-0
• Degree: $1$
• Conductor: $319$
• Sign: $0.970 - 0.242i$
• Analytic cond.: $34.2813$
• Root an. cond.: $34.2813$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$319$    =    $11 \cdot 29$
Sign:	$0.970 - 0.242i$
Analytic conductor:	$34.2813$
Root analytic conductor:	$34.2813$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{319} (173, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 319,\ (1:\ ),\ 0.970 - 0.242i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$2.975812655 - 0.3658061893i$
$L(1)$	$\approx$	$1.585550907 - 0.5275101745i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−25.00018560080073000000176230328, −24.306406343194257486165057012, −23.392626856524677192394155702468, −22.2987797747203613578158649618, −21.28904085569709369363347263663, −20.70433788863753874033022238263, −19.88441200151556189827417505339, −18.508539427728648530760606009489, −17.29066905771034104004073952698, −16.84943600715721988968165194505, −15.74258844098168473458484634828, −15.013150509723726162574867132924, −14.0876393655680954639106170840, −13.36679869959395757050195681919, −12.530041709766043236432171576451, −10.9364926901540191355962345337, −9.66636548552877858481562000924, −8.90031573341954057788673883268, −8.00135781345368706552250618979, −7.28435294987115872374017086993, −5.547394137932033107170402708168, −4.80439807361477790875663226978, −4.03256451482309436320013143031, −2.58755174232313934436942576403, −0.73553157410178355830529433022, 1.50543301574464552436575473878, 2.195530317694804362318169512985, 3.19518970913692417604369423107, 4.32599980553441444912526757487, 5.78245112027034053327127762257, 6.853058092233154513142582873202, 8.20181478924724043863194536269, 9.040794128422663353874912289876, 10.10289509690587626039836255395, 11.117013288041133460591511119263, 12.0386931811889496066025056807, 12.93535174945673665263176478249, 13.98917758257257759494122148251, 14.67018353725651037384893309972, 15.08990875531267449428940106538, 17.2048181244766172643515620126, 18.11717662501245453976940870396, 18.91361086703762068659227453703, 19.262527712428900111134783211180, 20.52480665296935016954944766415, 21.442391633503382261712779359624, 21.75656437964906827838199653268, 23.211859384497717395086702592458, 23.809300649567949419639199788738, 24.89170904454863970285847357285

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