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Properties

Label 1-11e2-121.69-r0-0-0
Degree $1$
Conductor $121$
Sign $-0.708 + 0.706i$
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  + (−0.921 + 0.389i)2-s + (−0.809 + 0.587i)3-s + (0.696 − 0.717i)4-s + (−0.254 + 0.967i)5-s + (0.516 − 0.856i)6-s + (0.941 + 0.336i)7-s + (−0.362 + 0.931i)8-s + (0.309 − 0.951i)9-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)12-s + (0.897 + 0.441i)13-s + (−0.998 + 0.0570i)14-s + (−0.362 − 0.931i)15-s + (−0.0285 − 0.999i)16-s + (−0.736 + 0.676i)17-s + (0.0855 + 0.996i)18-s + ⋯
L(s)  = 1  + (−0.921 + 0.389i)2-s + (−0.809 + 0.587i)3-s + (0.696 − 0.717i)4-s + (−0.254 + 0.967i)5-s + (0.516 − 0.856i)6-s + (0.941 + 0.336i)7-s + (−0.362 + 0.931i)8-s + (0.309 − 0.951i)9-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)12-s + (0.897 + 0.441i)13-s + (−0.998 + 0.0570i)14-s + (−0.362 − 0.931i)15-s + (−0.0285 − 0.999i)16-s + (−0.736 + 0.676i)17-s + (0.0855 + 0.996i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.708 + 0.706i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.708 + 0.706i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1932351331 + 0.4673686919i\)
\(L(\frac12)\) \(\approx\) \(0.1932351331 + 0.4673686919i\)
\(L(1)\) \(\approx\) \(0.4612629887 + 0.3369334112i\)
\(L(1)\) \(\approx\) \(0.4612629887 + 0.3369334112i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.921 + 0.389i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.254 + 0.967i)T \)
7 \( 1 + (0.941 + 0.336i)T \)
13 \( 1 + (0.897 + 0.441i)T \)
17 \( 1 + (-0.736 + 0.676i)T \)
19 \( 1 + (0.198 + 0.980i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (-0.870 + 0.491i)T \)
31 \( 1 + (-0.985 - 0.170i)T \)
37 \( 1 + (-0.466 + 0.884i)T \)
41 \( 1 + (0.974 + 0.226i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.0855 - 0.996i)T \)
53 \( 1 + (-0.0285 + 0.999i)T \)
59 \( 1 + (0.974 - 0.226i)T \)
61 \( 1 + (-0.921 - 0.389i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.993 + 0.113i)T \)
73 \( 1 + (0.610 + 0.791i)T \)
79 \( 1 + (0.774 - 0.633i)T \)
83 \( 1 + (-0.564 - 0.825i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.254 - 0.967i)T \)
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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

−28.40050290498239687721619949888, −27.92142291942389850758621932297, −27.07208995652040763135518523645, −25.65585530151058381099560797382, −24.40564298074240844082321395681, −24.05282773178791451790800164562, −22.59092789814877841905465605817, −21.242650596506895520384925991503, −20.34533519311228506661899835110, −19.44171943238645502699228646504, −17.95761477519117244442638198606, −17.70391735799398353757958270042, −16.46870017200901985238628583223, −15.70346009522532184013821247138, −13.55633921363147313741969214137, −12.58838080460774859115975386479, −11.432611032744238490413817378106, −10.9216197874605945744546979140, −9.24834114225403571272325200841, −8.09875681700026930918405377987, −7.25133227904168566163209290854, −5.64558756787515785331625966266, −4.216883165395565812270994723823, −1.95508961793995449784921131693, −0.73395664596134848507534692301, 1.82634204411086144773365863019, 3.9230532576510951559611670450, 5.62205677432233550116172496089, 6.51186124185613961975917063611, 7.83800788353144073347389292666, 9.07870027650685218903615884876, 10.47551795841018673369747277111, 11.0515499100789941312220844517, 11.96595304765338365359575488816, 14.28062580425562121446047294253, 15.13509758243648751419600033104, 16.00645610687338286146806853466, 17.11907023636636473607400824262, 18.20039488006493834067934061685, 18.590874013900012960812915127684, 20.214374719602595670701284506445, 21.27969226202867812908175085462, 22.374899027915883795777094175879, 23.542307892697010439730630179466, 24.266955519830098660254255296213, 25.79236432529473286166729469854, 26.53520503646078925381343583231, 27.4725669535088552692039012391, 28.07381762762063347285980043067, 29.14302628267154423089067746676

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