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Properties

Label 1-111-111.68-r0-0-0
Degree $1$
Conductor $111$
Sign $0.763 - 0.646i$
Analytic cond. $0.515481$
Root an. cond. $0.515481$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + i·5-s + 7-s + i·8-s + 10-s + 11-s − i·13-s − i·14-s + 16-s + i·17-s − i·19-s − i·20-s − i·22-s + i·23-s + ⋯
L(s)  = 1  − i·2-s − 4-s + i·5-s + 7-s + i·8-s + 10-s + 11-s − i·13-s − i·14-s + 16-s + i·17-s − i·19-s − i·20-s − i·22-s + i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.763 - 0.646i$
Analytic conductor: \(0.515481\)
Root analytic conductor: \(0.515481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (0:\ ),\ 0.763 - 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9881035017 - 0.3622677591i\)
\(L(\frac12)\) \(\approx\) \(0.9881035017 - 0.3622677591i\)
\(L(1)\) \(\approx\) \(1.005205997 - 0.3261861599i\)
\(L(1)\) \(\approx\) \(1.005205997 - 0.3261861599i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 \)
good2 \( 1 \)
5 \( 1 - iT \)
7 \( 1 \)
11 \( 1 - T \)
13 \( 1 + iT \)
17 \( 1 \)
19 \( 1 + T \)
23 \( 1 + iT \)
29 \( 1 \)
31 \( 1 + T \)
41 \( 1 \)
43 \( 1 - iT \)
47 \( 1 - iT \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 \)
83 \( 1 - iT \)
89 \( 1 + iT \)
97 \( 1 \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.537090350716544791381195641623, −28.12436458621962665703466427156, −27.48179409806595153996735324015, −26.55622581463859429564152011341, −25.11055829143777148463840964884, −24.551109758737302616442485186118, −23.76250379675297104890885758263, −22.59509342469281790479679432973, −21.37418382078989191225808917594, −20.37507869049141348308132747817, −18.92717518187404463262340159150, −17.8145091671361845659028858432, −16.767852071157142961510899708689, −16.19418533316915232310880367659, −14.61908872242994616993331030442, −14.037225536500387966801553233653, −12.6173498942671991394491783829, −11.51769663487572904239078083469, −9.5744683738891522423469575518, −8.729210781987695480264678210541, −7.68783893911254707954066243435, −6.30617182653870183854456625765, −4.94835953615715578724879176005, −4.14870331785780168197330348642, −1.39957129056617761445608487221, 1.595431243656791443714747977606, 3.03592437113616164168518586034, 4.279001187352969531211547410750, 5.80436208548715236602973847442, 7.53497363543995876552634279051, 8.78371933860792514234624149618, 10.1770931808642285897416934515, 11.07911670319207838558988495661, 11.89542029520899863820477135500, 13.34612829546419814576021469075, 14.41888668486238679217372262260, 15.21026045571733586511896222171, 17.48633340862625147091893649035, 17.75109017990850829481342625104, 19.171517977172636717104438845574, 19.87747201181742139296132788829, 21.21194819607120774285783591695, 21.97315009033450599590816103648, 22.88516475180933603338049264181, 23.98528421877642107708251766347, 25.429559492161978341019745817681, 26.64510468541732578957644109433, 27.43323137913473421562073271683, 28.21248907353080012274514077345, 29.66440527821468153506502899863

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