Properties

Label 1-11e2-121.82-r0-0-0
Degree $1$
Conductor $121$
Sign $-0.981 + 0.190i$
Analytic cond. $0.561921$
Root an. cond. $0.561921$
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  + (0.198 + 0.980i)2-s + (0.309 + 0.951i)3-s + (−0.921 + 0.389i)4-s + (0.610 + 0.791i)5-s + (−0.870 + 0.491i)6-s + (−0.985 − 0.170i)7-s + (−0.564 − 0.825i)8-s + (−0.809 + 0.587i)9-s + (−0.654 + 0.755i)10-s + (−0.654 − 0.755i)12-s + (0.974 + 0.226i)13-s + (−0.0285 − 0.999i)14-s + (−0.564 + 0.825i)15-s + (0.696 − 0.717i)16-s + (−0.362 − 0.931i)17-s + (−0.736 − 0.676i)18-s + ⋯
L(s)  = 1  + (0.198 + 0.980i)2-s + (0.309 + 0.951i)3-s + (−0.921 + 0.389i)4-s + (0.610 + 0.791i)5-s + (−0.870 + 0.491i)6-s + (−0.985 − 0.170i)7-s + (−0.564 − 0.825i)8-s + (−0.809 + 0.587i)9-s + (−0.654 + 0.755i)10-s + (−0.654 − 0.755i)12-s + (0.974 + 0.226i)13-s + (−0.0285 − 0.999i)14-s + (−0.564 + 0.825i)15-s + (0.696 − 0.717i)16-s + (−0.362 − 0.931i)17-s + (−0.736 − 0.676i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.981 + 0.190i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.981 + 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1056420247 + 1.096305617i\)
\(L(\frac12)\) \(\approx\) \(0.1056420247 + 1.096305617i\)
\(L(1)\) \(\approx\) \(0.6267741078 + 0.9182626619i\)
\(L(1)\) \(\approx\) \(0.6267741078 + 0.9182626619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.198 + 0.980i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.610 + 0.791i)T \)
7 \( 1 + (-0.985 - 0.170i)T \)
13 \( 1 + (0.974 + 0.226i)T \)
17 \( 1 + (-0.362 - 0.931i)T \)
19 \( 1 + (0.774 + 0.633i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (-0.254 - 0.967i)T \)
31 \( 1 + (0.0855 - 0.996i)T \)
37 \( 1 + (0.516 + 0.856i)T \)
41 \( 1 + (0.993 + 0.113i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.736 + 0.676i)T \)
53 \( 1 + (0.696 + 0.717i)T \)
59 \( 1 + (0.993 - 0.113i)T \)
61 \( 1 + (0.198 - 0.980i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (-0.998 - 0.0570i)T \)
73 \( 1 + (0.897 + 0.441i)T \)
79 \( 1 + (0.941 - 0.336i)T \)
83 \( 1 + (-0.466 + 0.884i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.610 - 0.791i)T \)
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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

−28.56221336850857984052926085971, −28.32239259118635120896434086354, −26.44157556663042177278185531109, −25.549948330384991428293442311160, −24.42644653174181278089836313892, −23.470873774338705471577348432, −22.42516712263067728939032458592, −21.298772555101241387344578544469, −20.16423008731541115556114775475, −19.615372050476707991971104981815, −18.38920162241068618558601990146, −17.67705658711838556738906835633, −16.2406109838270667155499950562, −14.53015629824006819703671704192, −13.263903351248978582044649267159, −12.99414438360795993426929072192, −11.95216616835019248600656777431, −10.479482556826801083730023745500, −9.14457995099231886150530682482, −8.49600135314135510963183983754, −6.50350387057239168879465408396, −5.44241607701724313145420460343, −3.6442180539286705491457862158, −2.3514953207279535427201045198, −1.02232244809605803741882756109, 2.99145635775782961499159005251, 3.96596045642769890694325911620, 5.56303598156062946788888015572, 6.465323649434942137031133522337, 7.83179291113053980916321045448, 9.406547374947761116945711687717, 9.82232302511718276933601524675, 11.37744307185866613720344163889, 13.40415135552807340745073885987, 13.87787241500463379929025479369, 15.117597787004451579276803482573, 15.94052649111005606557168491699, 16.78319777658412153615211098924, 18.04331604033058635127675630136, 19.07088884361077028665553088105, 20.60921559255344974829579861378, 21.694444772342922337443671443895, 22.567678557181211702316340694450, 23.09784141033285852586556616874, 24.85077007280285356124447783119, 25.69229754409516528131341700002, 26.27586844805141337851720730979, 27.07054127298937471505367084296, 28.30332769529479039874171828639, 29.54131589552531211835566201117

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