Properties

Degree $1$
Conductor $121$
Sign $-0.457 + 0.889i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.457 + 0.889i$
Motivic weight: \(0\)
Character: $\chi_{121} (115, \cdot )$
Sato-Tate group: $\mu(55)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.457 + 0.889i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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