L-function 1-10e2-100.31-r1-0-0

• Label: 1-10e2-100.31-r1-0-0
• Degree: $1$
• Conductor: $100$
• Sign: $0.0627 - 0.998i$
• Analytic cond.: $10.7464$
• Root an. cond.: $10.7464$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign:	$0.0627 - 0.998i$
Analytic conductor:	\(10.7464\)
Root analytic conductor:	\(10.7464\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{100} (31, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 100,\ (1:\ ),\ 0.0627 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.4608449620 - 0.4327622248i\)
\(L(1)\)	\(\approx\)	\(0.7371752092 + 0.04637913299i\)

Euler product

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

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

−29.563502566082643705808681646508, −29.283621861108947538963220336918, −28.05261319418904798998908232915, −26.870030724653214897204591634118, −25.3636196453880871767195300260, −25.01592319325559569953762120866, −23.55595774476149595621941747832, −22.8173372724719618114590859359, −21.8442014634616648246486865595, −20.145323022743450685270848639687, −19.32141236736612271782480742136, −18.42184684644512107039973227453, −17.133198212650653751092867350160, −16.37978554724940669560305561433, −14.7185416177557305374089236413, −13.634274746232372565858670022206, −12.43666431849570842358704151760, −11.78635466790946841521428118965, −10.11484623580118607695810894943, −8.91173398224218631146361475032, −7.236485812334507357772093424121, −6.59628993716829020219631557170, −5.08834724351264668510780704642, −3.18456918364470671829442409841, −1.560246336727401102574530402334, 0.29022636437210361807260763837, 2.947463480370356448961343047482, 4.126720332424636467799032055066, 5.60256557437483113883434165820, 6.72057165798615857117706644475, 8.61989299945966259778032793596, 9.67278039225319498766397784975, 10.64388593886479702599395688675, 11.901419066263659124349666009514, 13.12932630035721949013112148602, 14.64983897951443198065927235051, 15.52061725834967597949192436065, 16.71975689019295802904888902178, 17.33313305804766079163347434630, 19.133362088459581217234344806444, 19.911056445207523374333679212651, 21.26112397752685396406947392780, 22.19172881622086970056046564509, 22.83439544703337502566509660660, 24.22175256779656038387769883602, 25.5203691062573707028527624533, 26.45035001564266570915879399980, 27.3660442653951492860851023994, 28.3652119654551698465173174414, 29.2812328568194312208501732734

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