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Properties

Label 1-11e2-121.37-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: $\chi_{121} (37, \cdot )$
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

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

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