L-function 1-200-200.3-r0-0-0

• Label: 1-200-200.3-r0-0-0
• Degree: $1$
• Conductor: $200$
• Sign: $-0.535 + 0.844i$
• Analytic cond.: $0.928796$
• Root an. cond.: $0.928796$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.951 + 0.309i)3^-s − i·7^-s + (0.809 − 0.587i)9^-s + (−0.809 − 0.587i)11^-s + (0.587 + 0.809i)13^-s + (−0.951 − 0.309i)17^-s + (−0.309 + 0.951i)19^-s + (−0.309 − 0.951i)21^-s + (−0.587 + 0.809i)23^-s + (−0.587 + 0.809i)27^-s + (0.309 + 0.951i)29^-s + (−0.309 + 0.951i)31^-s + (0.951 + 0.309i)33^-s + (−0.587 − 0.809i)37^-s + (−0.809 − 0.587i)39^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$200$    =    $2^{3} \cdot 5^{2}$
Sign:	$-0.535 + 0.844i$
Analytic conductor:	$0.928796$
Root analytic conductor:	$0.928796$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{200} (3, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 200,\ (0:\ ),\ -0.535 + 0.844i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.2736371921 + 0.4977442046i$
$L(1)$	$\approx$	$0.6348273731 + 0.2427883576i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
good	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	$1 + (0.309 + 0.951i)T$
	31	$1 + (-0.309 + 0.951i)T$

Imaginary part of the first few zeros on the critical line

−26.57761830965674449661441370819, −25.74369359865218948348348489465, −24.37913913773818130215215966317, −23.70509100498744900591390220544, −22.87121290779899380166009628323, −22.12754015394932319348922666802, −20.80816661388769053122250856843, −20.00332288235995144500754714064, −18.73853893207091299041504880736, −17.7405242142036787571914343579, −17.20284323095417001067093471076, −16.013596864777135202681543039, −15.18542306842532422590527125118, −13.42916856746294803756199608610, −13.08505048788713214039461574377, −11.74766324002514017239157462626, −10.66148189270874733899276812532, −10.15095431897031323610037113129, −8.33608436952742948545425502941, −7.24862552425086871583039428779, −6.34447372264862091572902872936, −5.05949758211529482372644394839, −4.05983410113788513272787845507, −2.17897379614529462617909654840, −0.48108368046527431566392026400, 1.78874638294315146385941756382, 3.48174431867222170946660562005, 4.893190964887223781508693277259, 5.803503988798860704150504073570, 6.74192696673321391780177836973, 8.3386215869800799182853315356, 9.36944738862950554579174417906, 10.603610416977542054993547511746, 11.460827543398259143184464612338, 12.34297709694154324886976961850, 13.43307794286949793219184981211, 14.80788992279846331046319307130, 16.01294580490254474812171517052, 16.27798990029838089765233598344, 17.879604653838630871607178078770, 18.31408714582767145100586525576, 19.42552065449808579392271934040, 21.025809971481356982169750566110, 21.512704934878231056255558880944, 22.42837521574886261922992895953, 23.48311238578441950553402236349, 24.1752712910537416998026975948, 25.31684479177583466617918906003, 26.42989172226968078956894030457, 27.3285530271797878155946541393

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