Properties

Degree $1$
Conductor $121$
Sign $-0.890 + 0.455i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.516 + 0.856i)2-s + (−0.809 + 0.587i)3-s + (−0.466 + 0.884i)4-s + (0.774 + 0.633i)5-s + (−0.921 − 0.389i)6-s + (0.610 + 0.791i)7-s + (−0.998 + 0.0570i)8-s + (0.309 − 0.951i)9-s + (−0.142 + 0.989i)10-s + (−0.142 − 0.989i)12-s + (−0.985 − 0.170i)13-s + (−0.362 + 0.931i)14-s + (−0.998 − 0.0570i)15-s + (−0.564 − 0.825i)16-s + (0.993 + 0.113i)17-s + (0.974 − 0.226i)18-s + ⋯
L(s,χ)  = 1  + (0.516 + 0.856i)2-s + (−0.809 + 0.587i)3-s + (−0.466 + 0.884i)4-s + (0.774 + 0.633i)5-s + (−0.921 − 0.389i)6-s + (0.610 + 0.791i)7-s + (−0.998 + 0.0570i)8-s + (0.309 − 0.951i)9-s + (−0.142 + 0.989i)10-s + (−0.142 − 0.989i)12-s + (−0.985 − 0.170i)13-s + (−0.362 + 0.931i)14-s + (−0.998 − 0.0570i)15-s + (−0.564 − 0.825i)16-s + (0.993 + 0.113i)17-s + (0.974 − 0.226i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.890 + 0.455i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.890 + 0.455i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.890 + 0.455i$
Motivic weight: \(0\)
Character: $\chi_{121} (58, \cdot )$
Sato-Tate group: $\mu(55)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.890 + 0.455i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.2590323911 + 1.075870644i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.2590323911 + 1.075870644i\)
\(L(\chi,1)\) \(\approx\) \(0.7152244936 + 0.8529201721i\)
\(L(1,\chi)\) \(\approx\) \(0.7152244936 + 0.8529201721i\)

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

−28.84260896753168075303257826138, −27.944050386212295757243542721713, −27.089163542346764288934418113772, −25.26737674684046161728886054389, −24.11412389610944029906108307045, −23.65769459491343567592328932389, −22.441222518587272698931001147882, −21.5056072323831714005762958460, −20.60209279558470732736828385040, −19.50524906742581119719211477741, −18.33255934330447899538831612182, −17.363837914819222857688584799838, −16.5489303812488174467391615900, −14.52326189670636871636016957273, −13.70585120500652691312215824161, −12.63445345075022947978539981515, −11.9064707905847344247005948397, −10.589591849630372074267367891011, −9.82235488545565493455938364574, −8.04571643142942850600276866898, −6.412096457879471290859673622586, −5.25958965454257127312415088749, −4.377526880573134249493518006803, −2.19852827851604604566725145339, −1.06323328769036069638684127013, 2.62657440362468639547107482955, 4.35392446873998051813939076415, 5.5180251061251902732189717509, 6.17673235469320013660538958571, 7.57552588642362874773859755643, 9.13862170479111577180733054819, 10.25415636937740620812501886025, 11.68752738997366629385185216144, 12.60004348724765659619916706258, 14.18032514800317764410794693519, 14.92929034735944879806769351445, 15.84515499743640219140359598322, 17.26780476327915325337858022686, 17.595116148911067410989251893995, 18.79789476108746007260525842734, 21.025770287840455639964356596299, 21.62854450922217478826522677069, 22.309132909080350287969140480307, 23.32936043689857349223047185801, 24.40445194906085260233434946701, 25.37427634076485798901876571443, 26.38867493248758195143414616240, 27.330156745291804744810736003109, 28.30662939116243458998872716212, 29.6782574974450281212766827615

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