L-function 1-1100-1100.439-r0-0-0

• Label: 1-1100-1100.439-r0-0-0
• Degree: $1$
• Conductor: $1100$
• Sign: $-0.728 - 0.684i$
• Analytic cond.: $5.10837$
• Root an. cond.: $5.10837$
• 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 − 7^-s + (0.309 + 0.951i)9^-s + (0.309 + 0.951i)13^-s + (−0.809 + 0.587i)17^-s + (−0.809 + 0.587i)19^-s + (0.809 + 0.587i)21^-s + (0.309 − 0.951i)23^-s + (0.309 − 0.951i)27^-s + (0.809 + 0.587i)29^-s + (0.809 − 0.587i)31^-s + (−0.309 − 0.951i)37^-s + (0.309 − 0.951i)39^-s + (−0.309 − 0.951i)41^-s − 43^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign:	$-0.728 - 0.684i$
Analytic conductor:	\(5.10837\)
Root analytic conductor:	\(5.10837\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1100} (439, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 1100,\ (0:\ ),\ -0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.1338761477 - 0.3381325513i\)
\(L(1)\)	\(\approx\)	\(0.6146713991 - 0.1062859451i\)

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−21.70316208337879998584654654925, −21.161495107021629757888231082043, −20.07806303266633385140693861951, −19.53724809771050982814961534564, −18.45144181220663853159792347778, −17.6979949617983843317842756461, −17.06824971475950142637966851196, −16.19285985626083776323790286162, −15.48822529301269522939487935563, −15.128657696426251720606696867034, −13.6066538041825404303239353390, −13.08559716884996560923136348025, −12.14820199409503133737890047903, −11.3814521626693403449784273852, −10.528468190971287471020922603180, −9.8808563650929945919858911914, −9.1144811177147341644019886239, −8.12219527858649737507224147612, −6.71403681259635596483098293101, −6.43070420225385741019169822910, −5.3046061223990604554675857881, −4.57766740252604287441529395079, −3.49370344565326828247763932926, −2.736177548508928811115553546422, −1.0387307850226017505774426230, 0.195292040506789894794136573488, 1.61118677416169877577329379614, 2.51799407490127002502573776688, 3.865049019177261107865496941213, 4.6741271443588626038977782336, 5.87281907036291680560833433642, 6.543629246195152259235934454475, 6.98568731779572358053870274480, 8.286162562245565296085392769556, 9.01374022079871716975062752215, 10.23867954280998401607685968745, 10.7154348178238375612485650216, 11.778557494672407392071350643379, 12.42386500371501074975854675682, 13.15011118989102182998710562533, 13.806260703996701141213852349999, 14.905303257516482223179687892645, 15.924479193162285822510976489574, 16.533816659261009298500162721500, 17.146604899464257339567043571510, 18.04905802046113255902793453273, 18.921736813312783080390031895, 19.25714051134791974953904834537, 20.20296929944541046212488270665, 21.37904632754248127921933304613

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