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Properties

Label 1-11e2-121.20-r0-0-0
Degree $1$
Conductor $121$
Sign $-0.457 - 0.889i$
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  + (−0.362 − 0.931i)2-s + (0.309 − 0.951i)3-s + (−0.736 + 0.676i)4-s + (0.696 + 0.717i)5-s + (−0.998 + 0.0570i)6-s + (0.516 − 0.856i)7-s + (0.897 + 0.441i)8-s + (−0.809 − 0.587i)9-s + (0.415 − 0.909i)10-s + (0.415 + 0.909i)12-s + (0.198 − 0.980i)13-s + (−0.985 − 0.170i)14-s + (0.897 − 0.441i)15-s + (0.0855 − 0.996i)16-s + (0.610 − 0.791i)17-s + (−0.254 + 0.967i)18-s + ⋯
L(s)  = 1  + (−0.362 − 0.931i)2-s + (0.309 − 0.951i)3-s + (−0.736 + 0.676i)4-s + (0.696 + 0.717i)5-s + (−0.998 + 0.0570i)6-s + (0.516 − 0.856i)7-s + (0.897 + 0.441i)8-s + (−0.809 − 0.587i)9-s + (0.415 − 0.909i)10-s + (0.415 + 0.909i)12-s + (0.198 − 0.980i)13-s + (−0.985 − 0.170i)14-s + (0.897 − 0.441i)15-s + (0.0855 − 0.996i)16-s + (0.610 − 0.791i)17-s + (−0.254 + 0.967i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.457 - 0.889i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.457 - 0.889i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5320377641 - 0.8720137048i\)
\(L(\frac12)\) \(\approx\) \(0.5320377641 - 0.8720137048i\)
\(L(1)\) \(\approx\) \(0.7720707219 - 0.6583728028i\)
\(L(1)\) \(\approx\) \(0.7720707219 - 0.6583728028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.362 - 0.931i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.696 + 0.717i)T \)
7 \( 1 + (0.516 - 0.856i)T \)
13 \( 1 + (0.198 - 0.980i)T \)
17 \( 1 + (0.610 - 0.791i)T \)
19 \( 1 + (-0.564 + 0.825i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + (-0.0285 - 0.999i)T \)
31 \( 1 + (-0.870 + 0.491i)T \)
37 \( 1 + (0.993 + 0.113i)T \)
41 \( 1 + (0.774 - 0.633i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.254 - 0.967i)T \)
53 \( 1 + (0.0855 + 0.996i)T \)
59 \( 1 + (0.774 + 0.633i)T \)
61 \( 1 + (-0.362 + 0.931i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (0.941 - 0.336i)T \)
73 \( 1 + (-0.921 - 0.389i)T \)
79 \( 1 + (-0.466 + 0.884i)T \)
83 \( 1 + (0.974 + 0.226i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.696 - 0.717i)T \)
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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

−28.756687920057367214572422907870, −28.129303055684681903490117790127, −27.422788172208493786336125208526, −26.05218240245201537535251065030, −25.60777227817924951188561204300, −24.48850274832985440544759233467, −23.64792336061669906473628067756, −21.982723118127378462910423555027, −21.5019774782190821513729077748, −20.25049810338429783014274839452, −18.96272597231117020799407498721, −17.769518043845082368650610425, −16.74577368966082793512260241636, −16.04690278694232386869199723480, −14.8214560895265019593728384819, −14.18377016079763815560118597409, −12.79482744770884841429871503468, −11.02701471215528526383171595694, −9.70522288095534383301895069364, −8.92965198790989078200044411897, −8.18457756572871853901481677206, −6.22788019389022595320564935742, −5.23115554051795353128545639154, −4.26720350519793642238695472940, −1.97906036497666934846854949818, 1.22268804569264327917930213694, 2.4661855985307626224373298473, 3.68134534999211637244183292511, 5.68992038807700607850903240778, 7.32676443811782997772346165735, 8.09545961276603164313292591018, 9.63572060913863430535356928609, 10.63294700549299763502405585074, 11.698153519260574096664590416279, 12.986530963712354741746654466550, 13.79449623003394225345668814394, 14.611380063234435225300956911336, 16.85465171607879881512912034785, 17.84083482277370646408069424322, 18.338651238169629214932054222949, 19.49709785805555901411876698895, 20.4378845697532586575566926697, 21.28863112466341320677616154903, 22.71982440013276580042666659034, 23.36365330469535438573426822572, 24.98475249610403432471855391818, 25.75716834310287216273472811758, 26.75881503979359631600599515248, 27.74529419590623920492472588474, 29.1774005554031640034247700593

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