L-function 1-319-319.171-r0-0-0

• Label: 1-319-319.171-r0-0-0
• Degree: $1$
• Conductor: $319$
• Sign: $-0.289 - 0.957i$
• 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

+ (−0.880 − 0.473i)2^-s + (−0.834 − 0.550i)3^-s + (0.550 + 0.834i)4^-s + (−0.983 − 0.178i)5^-s + (0.473 + 0.880i)6^-s + (−0.936 + 0.351i)7^-s + (−0.0896 − 0.995i)8^-s + (0.393 + 0.919i)9^-s + (0.781 + 0.623i)10^-s − i·12^-s + (0.753 + 0.657i)13^-s + (0.990 + 0.134i)14^-s + (0.722 + 0.691i)15^-s + (−0.393 + 0.919i)16^-s + (−0.587 + 0.809i)17^-s + (0.0896 − 0.995i)18^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$319$    =    $11 \cdot 29$
Sign:	$-0.289 - 0.957i$
Analytic conductor:	$1.48142$
Root analytic conductor:	$1.48142$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{319} (171, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 319,\ (0:\ ),\ -0.289 - 0.957i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.1851185357 - 0.2494942141i$
$L(1)$	$\approx$	$0.3811982906 - 0.1418099143i$

Euler product

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

	$p$	$F_p(T)$
bad	11	$1$
	29	$1$
good	2	$1 + (-0.880 - 0.473i)T$
	3	$1 + (-0.834 - 0.550i)T$
	5	$1 + (-0.983 - 0.178i)T$
	7	$1 + (-0.936 + 0.351i)T$
	13	$1 + (0.753 + 0.657i)T$
	17	$1 + (-0.587 + 0.809i)T$
	19	$1 + (0.351 - 0.936i)T$
	23	$1 + (-0.900 - 0.433i)T$
	31	$1 + (0.880 + 0.473i)T$

Imaginary part of the first few zeros on the critical line

−25.68392249728663778873922425017, −24.509811680099733776409895657675, −23.48343738328638921515029380811, −22.95562239386160409752314868267, −22.21746506682864985997897121595, −20.54754290419922780540280691524, −20.09098040303207055412941942502, −18.86921943422028215763360560823, −18.240846421931287481477842248393, −17.161334177042522898493325340075, −16.22462201044483186026772159213, −15.80545500273976100629335984068, −15.07841536608454957142336681279, −13.63019014052057864462094136769, −12.1387339046153991294720711011, −11.3942158738363132844705076670, −10.40404879337376744052962662487, −9.778328405282804955279681524037, −8.553325721952438571708655877888, −7.45257228468068755959081284554, −6.52714705002593862883578107872, −5.663274608314923842470424170305, −4.272551990483358204454671368738, −3.13179881410007586908090887274, −0.899123734282260299277577385670, 0.411903932632879845573360756744, 1.88856565786682972086661668860, 3.32043476626717861016853063528, 4.500903638639459740805347937094, 6.27722815488221970668451486050, 6.87419741400716236033499385161, 8.06153289638971849781043755589, 8.89126310310792113350284939449, 10.14578099461702033595110671916, 11.16772299857637619566701670845, 11.80242391294496349440429274587, 12.63438342994104450265097463958, 13.406721986951264819606292223588, 15.42155611973267433725982404733, 16.13084462222320843335487693852, 16.72500040034241487365469669135, 17.920835323674628311662344155337, 18.5963303079925856351250936407, 19.50109353437525490121674369189, 19.8939968757461241885083602317, 21.389666035207480523349886783455, 22.1877999876274970537070639915, 23.10633340765143516962104328321, 24.040444565855750595449252637409, 24.854293920843292957282159312014

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