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Properties

Label 1-11e2-121.48-r0-0-0
Degree $1$
Conductor $121$
Sign $-0.890 - 0.455i$
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  + (0.516 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.890 - 0.455i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.890 - 0.455i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.516 - 0.856i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.774 - 0.633i)T \)
7 \( 1 + (0.610 - 0.791i)T \)
13 \( 1 + (-0.985 + 0.170i)T \)
17 \( 1 + (0.993 - 0.113i)T \)
19 \( 1 + (-0.870 + 0.491i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (0.198 + 0.980i)T \)
31 \( 1 + (0.897 - 0.441i)T \)
37 \( 1 + (0.696 + 0.717i)T \)
41 \( 1 + (0.0855 + 0.996i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.974 - 0.226i)T \)
53 \( 1 + (-0.564 - 0.825i)T \)
59 \( 1 + (0.0855 - 0.996i)T \)
61 \( 1 + (0.516 + 0.856i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (0.941 - 0.336i)T \)
79 \( 1 + (-0.254 + 0.967i)T \)
83 \( 1 + (-0.0285 + 0.999i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.774 + 0.633i)T \)
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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

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

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