L-function 1-5e2-25.22-r1-0-0

• Label: 1-5e2-25.22-r1-0-0
• Degree: $1$
• Conductor: $25$
• Sign: $0.248 - 0.968i$
• Analytic cond.: $2.68662$
• Root an. cond.: $2.68662$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.587 − 0.809i)2^-s + (0.951 − 0.309i)3^-s + (−0.309 − 0.951i)4^-s + (0.309 − 0.951i)6^-s + i·7^-s + (−0.951 − 0.309i)8^-s + (0.809 − 0.587i)9^-s + (−0.809 − 0.587i)11^-s + (−0.587 − 0.809i)12^-s + (0.587 + 0.809i)13^-s + (0.809 + 0.587i)14^-s + (−0.809 + 0.587i)16^-s + (0.951 + 0.309i)17^-s − i·18^-s + (−0.309 + 0.951i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$25$    =    $5^{2}$
Sign:	$0.248 - 0.968i$
Analytic conductor:	$2.68662$
Root analytic conductor:	$2.68662$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{25} (22, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 25,\ (1:\ ),\ 0.248 - 0.968i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.698892521 - 1.317796120i$
$L(1)$	$\approx$	$1.502935290 - 0.8146263744i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−38.59889367890269848499165209757, −36.74735205333305172005801626122, −35.975598991709876343890990784535, −34.27586574913220758342749273218, −32.97860683977653684130801577331, −32.24413557918853681821285672084, −30.8805355043744549490832883920, −29.98311919115548044573858749999, −27.560065896921924650537764698394, −26.23558771727709495506183551581, −25.53432201050028880350409263436, −24.01802416462035286251035286780, −22.80133874512205780707630217717, −21.14235793288864123145726798648, −20.15907046005617239439438131795, −18.092878207712702470918655031591, −16.396276979937542350697677652775, −15.203437904668665289310507087237, −13.927268770752639087339477676644, −12.86075528096509909418139352687, −10.281685234816200883568263210570, −8.36320321565733722042263037584, −7.162107251158793135539960620329, −4.82460368351358800972059216098, −3.233418702909588958935034996908, 2.00962811207997792106880625555, 3.60948880523282830801389596731, 5.85314952260678614135089002329, 8.35729297605591647017677515242, 9.83699969076060357130631425927, 11.787506531645565541452875646307, 13.1257342764403463460291410638, 14.34576141050023920624390795965, 15.64843473466794840508725022084, 18.53767676286194792550544945503, 19.10775540941618199150071974083, 20.829278426493166479653774278, 21.53611502144450401795286496697, 23.4153931595628825998142240338, 24.604934083227172991538570894799, 26.05638480343179985683091206165, 27.72683394011576383827045427617, 29.076146298482492391157443057720, 30.32285426605063366456286951633, 31.523862125357787500857504281564, 32.02657057510138798061776627674, 33.78789509266728952939908101014, 35.66072480695777225366142233666, 36.95296793427771887093297939565, 37.83914813124235530266748124924

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