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Properties

Label 1-11e2-121.36-r0-0-0
Degree $1$
Conductor $121$
Sign $-0.121 + 0.992i$
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  + (−0.736 − 0.676i)2-s + (−0.809 + 0.587i)3-s + (0.0855 + 0.996i)4-s + (−0.0285 − 0.999i)5-s + (0.993 + 0.113i)6-s + (−0.466 + 0.884i)7-s + (0.610 − 0.791i)8-s + (0.309 − 0.951i)9-s + (−0.654 + 0.755i)10-s + (−0.654 − 0.755i)12-s + (−0.921 + 0.389i)13-s + (0.941 − 0.336i)14-s + (0.610 + 0.791i)15-s + (−0.985 + 0.170i)16-s + (−0.254 + 0.967i)17-s + (−0.870 + 0.491i)18-s + ⋯
L(s)  = 1  + (−0.736 − 0.676i)2-s + (−0.809 + 0.587i)3-s + (0.0855 + 0.996i)4-s + (−0.0285 − 0.999i)5-s + (0.993 + 0.113i)6-s + (−0.466 + 0.884i)7-s + (0.610 − 0.791i)8-s + (0.309 − 0.951i)9-s + (−0.654 + 0.755i)10-s + (−0.654 − 0.755i)12-s + (−0.921 + 0.389i)13-s + (0.941 − 0.336i)14-s + (0.610 + 0.791i)15-s + (−0.985 + 0.170i)16-s + (−0.254 + 0.967i)17-s + (−0.870 + 0.491i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.121 + 0.992i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.121 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1858544786 + 0.2100397478i\)
\(L(\frac12)\) \(\approx\) \(0.1858544786 + 0.2100397478i\)
\(L(1)\) \(\approx\) \(0.4452132589 + 0.02653349381i\)
\(L(1)\) \(\approx\) \(0.4452132589 + 0.02653349381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.736 - 0.676i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.0285 - 0.999i)T \)
7 \( 1 + (-0.466 + 0.884i)T \)
13 \( 1 + (-0.921 + 0.389i)T \)
17 \( 1 + (-0.254 + 0.967i)T \)
19 \( 1 + (-0.362 + 0.931i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (-0.998 - 0.0570i)T \)
31 \( 1 + (0.516 + 0.856i)T \)
37 \( 1 + (0.974 - 0.226i)T \)
41 \( 1 + (0.198 + 0.980i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.870 - 0.491i)T \)
53 \( 1 + (-0.985 - 0.170i)T \)
59 \( 1 + (0.198 - 0.980i)T \)
61 \( 1 + (-0.736 + 0.676i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (0.696 - 0.717i)T \)
79 \( 1 + (-0.564 + 0.825i)T \)
83 \( 1 + (0.897 - 0.441i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.0285 + 0.999i)T \)
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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

−28.84437809458743081803667722552, −27.59026339866076125097109068604, −26.733271234249404388232977992, −25.88717562056598415839925764763, −24.708494923303609305663882624288, −23.81451464487774820947147648484, −22.79283913737662646566400777037, −22.26582866544957762565291472949, −20.143532528485578793657398385829, −19.1881610223791801437778760651, −18.33958890197712397815259945856, −17.41389705288416849340022858543, −16.63570122276147235327694124907, −15.44263639343864942337162517991, −14.2305567326436953646807715803, −13.15827238431836569070898712796, −11.45779674075550365562820670684, −10.595913782020259936604582985990, −9.66828221528643752679719896424, −7.73611013578107874840831085146, −7.02444956905142456357012036260, −6.23267105452840526962887328529, −4.741383592008198633703399385752, −2.42641461273245350207346439180, −0.358866797281209073368101199004, 1.73838998233365445542073771679, 3.642569457046507033093310017622, 4.9290022341600655360786144989, 6.29710834811767983637700116156, 8.093649640968744052578849162253, 9.30445421606619664121934638666, 9.91668661092425877564351535711, 11.37043697099115906363189977008, 12.25163450723997151403998495183, 12.918771968030948585014537682556, 15.13024481783817814534710331698, 16.27414957479103683426089616209, 16.91312494429275636417447565378, 17.885617641241757750112668988735, 19.1311496951712886651313257048, 20.086880686935346226644522650751, 21.4473415011483851905616823279, 21.67221290180188216452081438674, 23.03708127022636879052007667547, 24.38371650110708044838946263298, 25.45067444912536373935516565345, 26.669018058535217843883933772220, 27.59866445868304790607648219826, 28.387708710014653214405297311271, 28.89537744069838154868022900404

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