L-function 1-308-308.223-r0-0-0

• Label: 1-308-308.223-r0-0-0
• Degree: $1$
• Conductor: $308$
• Sign: $-0.822 - 0.568i$
• Analytic cond.: $1.43034$
• Root an. cond.: $1.43034$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.809 + 0.587i)3^-s + (−0.309 − 0.951i)5^-s + (0.309 − 0.951i)9^-s + (−0.309 + 0.951i)13^-s + (0.809 + 0.587i)15^-s + (−0.309 − 0.951i)17^-s + (−0.809 + 0.587i)19^-s − 23^-s + (−0.809 + 0.587i)25^-s + (0.309 + 0.951i)27^-s + (−0.809 − 0.587i)29^-s + (0.309 − 0.951i)31^-s + (−0.809 − 0.587i)37^-s + (−0.309 − 0.951i)39^-s + (0.809 − 0.587i)41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign:	$-0.822 - 0.568i$
Analytic conductor:	\(1.43034\)
Root analytic conductor:	\(1.43034\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{308} (223, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 308,\ (0:\ ),\ -0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.07227067841 - 0.2317647475i\)
\(L(1)\)	\(\approx\)	\(0.5799591020 - 0.05104825899i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	7	\( 1 \)
	11	\( 1 \)
good	3	\( 1 + (-0.809 + 0.587i)T \)
	5	\( 1 + (-0.309 - 0.951i)T \)
	13	\( 1 + (-0.309 + 0.951i)T \)
	17	\( 1 + (-0.309 - 0.951i)T \)
	19	\( 1 + (-0.809 + 0.587i)T \)
	23	\( 1 - T \)
	29	\( 1 + (-0.809 - 0.587i)T \)
	31	\( 1 + (0.309 - 0.951i)T \)
	37	\( 1 + (-0.809 - 0.587i)T \)

Imaginary part of the first few zeros on the critical line

−25.65612709493920517370190673036, −24.58565240546243158290056850755, −23.72358840484938383737044589645, −22.995336825311992397668948364170, −22.167339803544917768816069841442, −21.55871705869915560911543173119, −19.89931898918920332133316455754, −19.31522579694990012611384442579, −18.23369466792653659898734180174, −17.70621349460551681240104354239, −16.72535783451016945844248356703, −15.552990786300042783724096082790, −14.77822737260943587794484263332, −13.562425208787573595987657566858, −12.64565303274242593420948465889, −11.74653818467986615780629190810, −10.723521281408551296473064970552, −10.23728680730680028174575358622, −8.44473774764182607559810899586, −7.486948824105569733571593839979, −6.58889237577995712359635431076, −5.746960419678808260861674642095, −4.440818028261188311624023939311, −3.04218788297731916851010772617, −1.76611216691672381635299884120, 0.16733890167962566418896811946, 1.88482683847848500990410569648, 3.85366349887086904176647513665, 4.557544935015172141852964438132, 5.55542804454039835950261073616, 6.62895027362622426211887114859, 7.93572545589152966570838993849, 9.16738696622727076914659633788, 9.81533576778262058026497357527, 11.14115597048923478443469715139, 11.88209934753796588397482336631, 12.65383842524029536807441675947, 13.876707283331009883936925605871, 15.10028503275121233415114128932, 16.04020036298826075089965023397, 16.647263635467249240404466302600, 17.4069568296416638697049265340, 18.52160934142274249566801177661, 19.596006150897185312020143326654, 20.711450590863593378593606758088, 21.20395682894652892547920029822, 22.312435964274756903702676824988, 23.0940461150281976845373073678, 24.039335322669045648047679657266, 24.5966177061335378757219840459

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