L-function 1-111-111.110-r1-0-0

• Label: 1-111-111.110-r1-0-0
• Degree: $1$
• Conductor: $111$
• Sign: $1$
• Analytic cond.: $11.9286$
• Root an. cond.: $11.9286$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: yes
• Primitive: yes
• Self-dual: yes
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ 2^-s + 4^-s + 5^-s + 7^-s + 8^-s + 10^-s − 11^-s − 13^-s + 14^-s + 16^-s + 17^-s − 19^-s + 20^-s − 22^-s + 23^-s + 25^-s − 26^-s + 28^-s + 29^-s − 31^-s + 32^-s + 34^-s + 35^-s − 38^-s + 40^-s − 41^-s − 43^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(111\)    =    \(3 \cdot 37\)
Sign:	$1$
Analytic conductor:	\(11.9286\)
Root analytic conductor:	\(11.9286\)
Motivic weight:	\(0\)
Rational:	yes
Arithmetic:	yes
Character:	$\chi_{111} (110, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	\(0\)
Selberg data:	\((1,\ 111,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(4.018487349\)
\(L(1)\)	\(\approx\)	\(2.385494229\)

Euler product

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

	$p$	$F_p(T)$
bad	3	\( 1 \)
	37	\( 1 \)
good	2	\( 1 + T \)
	5	\( 1 + T \)
	7	\( 1 + T \)
	11	\( 1 - T \)
	13	\( 1 - T \)
	17	\( 1 + T \)
	19	\( 1 - T \)
	23	\( 1 + T \)
	29	\( 1 + T \)

Imaginary part of the first few zeros on the critical line

−29.410783343940605735060334563732, −28.55044247075892998197504751996, −27.153648614558493521094926481049, −25.75995499611335003359892012389, −24.993543275966750753966435287570, −23.99151932841177520078080327008, −23.13634028367934168997612094373, −21.67013488419763618327894787851, −21.281649104546924705742566477586, −20.32450705058597914708587668926, −18.824051675795694853728087854558, −17.49579586395880586962990542039, −16.602485150507739228363434008960, −15.03710621653526779194013713608, −14.392261552192090746786738833462, −13.26982252916463142779127787459, −12.32527336261593817132751592862, −10.94954657241707780180907292524, −10.02466677956892528031856770206, −8.17757446396435144676603620098, −6.90522776954867651394444893962, −5.44443005958836874755376360026, −4.80339932893683508528076265259, −2.86468766368800305685819078989, −1.71181386932933385219773230284, 1.71181386932933385219773230284, 2.86468766368800305685819078989, 4.80339932893683508528076265259, 5.44443005958836874755376360026, 6.90522776954867651394444893962, 8.17757446396435144676603620098, 10.02466677956892528031856770206, 10.94954657241707780180907292524, 12.32527336261593817132751592862, 13.26982252916463142779127787459, 14.392261552192090746786738833462, 15.03710621653526779194013713608, 16.602485150507739228363434008960, 17.49579586395880586962990542039, 18.824051675795694853728087854558, 20.32450705058597914708587668926, 21.281649104546924705742566477586, 21.67013488419763618327894787851, 23.13634028367934168997612094373, 23.99151932841177520078080327008, 24.993543275966750753966435287570, 25.75995499611335003359892012389, 27.153648614558493521094926481049, 28.55044247075892998197504751996, 29.410783343940605735060334563732

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