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Properties

Label 1-11e2-121.16-r0-0-0
Degree $1$
Conductor $121$
Sign $0.452 + 0.891i$
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  + (0.974 + 0.226i)2-s + (0.309 + 0.951i)3-s + (0.897 + 0.441i)4-s + (−0.362 − 0.931i)5-s + (0.0855 + 0.996i)6-s + (−0.0285 + 0.999i)7-s + (0.774 + 0.633i)8-s + (−0.809 + 0.587i)9-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)12-s + (−0.466 − 0.884i)13-s + (−0.254 + 0.967i)14-s + (0.774 − 0.633i)15-s + (0.610 + 0.791i)16-s + (0.198 − 0.980i)17-s + (−0.921 + 0.389i)18-s + ⋯
L(s)  = 1  + (0.974 + 0.226i)2-s + (0.309 + 0.951i)3-s + (0.897 + 0.441i)4-s + (−0.362 − 0.931i)5-s + (0.0855 + 0.996i)6-s + (−0.0285 + 0.999i)7-s + (0.774 + 0.633i)8-s + (−0.809 + 0.587i)9-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)12-s + (−0.466 − 0.884i)13-s + (−0.254 + 0.967i)14-s + (0.774 − 0.633i)15-s + (0.610 + 0.791i)16-s + (0.198 − 0.980i)17-s + (−0.921 + 0.389i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.452 + 0.891i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.452 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.590925218 + 0.9763412325i\)
\(L(\frac12)\) \(\approx\) \(1.590925218 + 0.9763412325i\)
\(L(1)\) \(\approx\) \(1.633880092 + 0.6589719796i\)
\(L(1)\) \(\approx\) \(1.633880092 + 0.6589719796i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.974 + 0.226i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.362 - 0.931i)T \)
7 \( 1 + (-0.0285 + 0.999i)T \)
13 \( 1 + (-0.466 - 0.884i)T \)
17 \( 1 + (0.198 - 0.980i)T \)
19 \( 1 + (0.993 + 0.113i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (-0.736 - 0.676i)T \)
31 \( 1 + (0.696 + 0.717i)T \)
37 \( 1 + (-0.985 - 0.170i)T \)
41 \( 1 + (0.516 - 0.856i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.921 - 0.389i)T \)
53 \( 1 + (0.610 - 0.791i)T \)
59 \( 1 + (0.516 + 0.856i)T \)
61 \( 1 + (0.974 - 0.226i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (-0.564 + 0.825i)T \)
79 \( 1 + (-0.998 + 0.0570i)T \)
83 \( 1 + (0.941 + 0.336i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.362 + 0.931i)T \)
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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

−29.537890862282902234439588836486, −28.197900068117867965128682313997, −26.41902745880920580574458754708, −25.91527400410612413614164340670, −24.37968124103982834650229523707, −23.80136426528986774164346115610, −22.92200535334012295462026185975, −21.99560379350711256814898759323, −20.60849911646632223383389714221, −19.61430215002052489012977712334, −19.02303718968012255794669958516, −17.63795466836694599997553561659, −16.25758214141631145230713259771, −14.77779857390347547430703648283, −14.15256140575106914283689085956, −13.27777551768776972504428737444, −12.02240883199342814388454710560, −11.18353526503213553165901078284, −9.925322350997131163710117208807, −7.755196942516964407038586666371, −7.03968454032971040132506809988, −6.04015468596520112373050405514, −4.12895403519021443399227265350, −3.07509596693788125997686139622, −1.66351302652423002446199028404, 2.4950623750075286278337923498, 3.72258429236354842900252086504, 5.05370231466597158415711865623, 5.58880723740519454867690025493, 7.66340718734448911248645087157, 8.71274221917086619643872808721, 9.98880036104841100664139709228, 11.6009249888650047109096390966, 12.30074632594065311226110622285, 13.62471948791070942659268636806, 14.758803467887333665287191801738, 15.79661829372890782761821626836, 16.150683694641990454301707409525, 17.54209479666320904978473512281, 19.45413242827306919813397734496, 20.46258634549488803773233997509, 21.037845958993551368029750592, 22.244493716463276047788403937495, 22.809496325211768120598181346734, 24.45161601888010682357006691698, 24.87677361521979805902384360422, 26.011509305965208642249220983180, 27.30408660352505292073871513238, 28.206654642051144515838384757350, 29.15336517023864544365682080710

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