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Properties

Label 1-11e2-121.4-r0-0-0
Degree $1$
Conductor $121$
Sign $0.906 + 0.422i$
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  + (0.993 + 0.113i)2-s + (−0.809 − 0.587i)3-s + (0.974 + 0.226i)4-s + (−0.564 + 0.825i)5-s + (−0.736 − 0.676i)6-s + (0.696 + 0.717i)7-s + (0.941 + 0.336i)8-s + (0.309 + 0.951i)9-s + (−0.654 + 0.755i)10-s + (−0.654 − 0.755i)12-s + (0.516 − 0.856i)13-s + (0.610 + 0.791i)14-s + (0.941 − 0.336i)15-s + (0.897 + 0.441i)16-s + (0.774 − 0.633i)17-s + (0.198 + 0.980i)18-s + ⋯
L(s)  = 1  + (0.993 + 0.113i)2-s + (−0.809 − 0.587i)3-s + (0.974 + 0.226i)4-s + (−0.564 + 0.825i)5-s + (−0.736 − 0.676i)6-s + (0.696 + 0.717i)7-s + (0.941 + 0.336i)8-s + (0.309 + 0.951i)9-s + (−0.654 + 0.755i)10-s + (−0.654 − 0.755i)12-s + (0.516 − 0.856i)13-s + (0.610 + 0.791i)14-s + (0.941 − 0.336i)15-s + (0.897 + 0.441i)16-s + (0.774 − 0.633i)17-s + (0.198 + 0.980i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.906 + 0.422i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.906 + 0.422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.455897615 + 0.3226545708i\)
\(L(\frac12)\) \(\approx\) \(1.455897615 + 0.3226545708i\)
\(L(1)\) \(\approx\) \(1.431457533 + 0.1667266486i\)
\(L(1)\) \(\approx\) \(1.431457533 + 0.1667266486i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.993 + 0.113i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.564 + 0.825i)T \)
7 \( 1 + (0.696 + 0.717i)T \)
13 \( 1 + (0.516 - 0.856i)T \)
17 \( 1 + (0.774 - 0.633i)T \)
19 \( 1 + (-0.998 - 0.0570i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (-0.362 + 0.931i)T \)
31 \( 1 + (-0.921 - 0.389i)T \)
37 \( 1 + (0.0855 - 0.996i)T \)
41 \( 1 + (-0.870 + 0.491i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (0.198 - 0.980i)T \)
53 \( 1 + (0.897 - 0.441i)T \)
59 \( 1 + (-0.870 - 0.491i)T \)
61 \( 1 + (0.993 - 0.113i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (-0.254 - 0.967i)T \)
73 \( 1 + (-0.466 - 0.884i)T \)
79 \( 1 + (-0.0285 - 0.999i)T \)
83 \( 1 + (-0.985 - 0.170i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.564 - 0.825i)T \)
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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

−28.91570788746545593237843958479, −28.14328345214358340149894324066, −27.25599883328001231908110280256, −25.94422604871555804029229121601, −24.39408868980893673853512360786, −23.598847047430023203946185030116, −23.20584251567565700103952556657, −21.789980833650867571746080341148, −20.93719623492987710514250380424, −20.322388427737851184819808313367, −18.886370146416626654911292136779, −16.98577963492755504156110497633, −16.60846331512477543912312824634, −15.416943311898240048848365759621, −14.430385775169268030431578146450, −13.04474403625137995541628854540, −12.01112664470306226474077583345, −11.19473537060337497348715630800, −10.201511477919596672411085057516, −8.40012723086751174012263782655, −6.890568035001585105546477481873, −5.595141166928866526429835178182, −4.4234359415965556511769077894, −3.89927081113573228800603215631, −1.412516001911912128869413385, 1.95838332971877893117942955178, 3.43712382674319464776757597644, 5.07174502086520452071035961238, 5.96567227307296710175167379855, 7.18791663835850096517596865580, 8.07752197126628256817314699270, 10.58108765325954270445311808551, 11.37712966652571866100359769472, 12.16566215241851642654940721309, 13.260751349151525045176894361330, 14.55959254637106231793562307090, 15.397696695502788925014064723979, 16.50682787613029403345194349414, 17.88441715299866506322789220745, 18.72659370378102298067394007530, 19.96834518265792570091233504649, 21.413104069373054506486293258312, 22.172312273507667344541260251924, 23.21607003394341073139705053643, 23.66861016515767534512415708191, 24.919124340611338766154058597491, 25.64443141510926744300747544141, 27.42101096140676865135579026683, 28.12608716758068700015498130300, 29.650429185573356335399194490436

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