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Properties

Label 1-11e2-121.49-r0-0-0
Degree $1$
Conductor $121$
Sign $-0.715 + 0.698i$
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  + (0.696 + 0.717i)2-s + (0.309 + 0.951i)3-s + (−0.0285 + 0.999i)4-s + (−0.870 + 0.491i)5-s + (−0.466 + 0.884i)6-s + (0.774 − 0.633i)7-s + (−0.736 + 0.676i)8-s + (−0.809 + 0.587i)9-s + (−0.959 − 0.281i)10-s + (−0.959 + 0.281i)12-s + (0.610 − 0.791i)13-s + (0.993 + 0.113i)14-s + (−0.736 − 0.676i)15-s + (−0.998 − 0.0570i)16-s + (0.0855 + 0.996i)17-s + (−0.985 − 0.170i)18-s + ⋯
L(s)  = 1  + (0.696 + 0.717i)2-s + (0.309 + 0.951i)3-s + (−0.0285 + 0.999i)4-s + (−0.870 + 0.491i)5-s + (−0.466 + 0.884i)6-s + (0.774 − 0.633i)7-s + (−0.736 + 0.676i)8-s + (−0.809 + 0.587i)9-s + (−0.959 − 0.281i)10-s + (−0.959 + 0.281i)12-s + (0.610 − 0.791i)13-s + (0.993 + 0.113i)14-s + (−0.736 − 0.676i)15-s + (−0.998 − 0.0570i)16-s + (0.0855 + 0.996i)17-s + (−0.985 − 0.170i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.715 + 0.698i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.715 + 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5464743280 + 1.341416872i\)
\(L(\frac12)\) \(\approx\) \(0.5464743280 + 1.341416872i\)
\(L(1)\) \(\approx\) \(0.9803593453 + 1.015041050i\)
\(L(1)\) \(\approx\) \(0.9803593453 + 1.015041050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.696 + 0.717i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.870 + 0.491i)T \)
7 \( 1 + (0.774 - 0.633i)T \)
13 \( 1 + (0.610 - 0.791i)T \)
17 \( 1 + (0.0855 + 0.996i)T \)
19 \( 1 + (-0.921 - 0.389i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (0.516 + 0.856i)T \)
31 \( 1 + (0.941 - 0.336i)T \)
37 \( 1 + (-0.564 + 0.825i)T \)
41 \( 1 + (0.897 - 0.441i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (-0.985 + 0.170i)T \)
53 \( 1 + (-0.998 + 0.0570i)T \)
59 \( 1 + (0.897 + 0.441i)T \)
61 \( 1 + (0.696 - 0.717i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (0.974 - 0.226i)T \)
73 \( 1 + (-0.254 - 0.967i)T \)
79 \( 1 + (0.198 + 0.980i)T \)
83 \( 1 + (-0.362 - 0.931i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.870 - 0.491i)T \)
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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

−28.75733872968291651960067993118, −27.9958944856275759173040684930, −26.93364586332417422028443160443, −25.15447206059936405277795759775, −24.444523226417435565332596636540, −23.51415827384495643188714099673, −22.91118451415154979891441508040, −21.1847828982714321325759374371, −20.65505171731521763996790885228, −19.29899550379123294622219347828, −18.90313681485237201189840734983, −17.647868322405751014319995124246, −15.890058326478350049793083290095, −14.79233659654052150884251526875, −13.84078691583283611270091275374, −12.71559032344765998440402343435, −11.816398208547814762532825810634, −11.2203560293089742012124868678, −9.17569060905851917508081429642, −8.2259467496161016997136296231, −6.765674131830843220509668934812, −5.339852119643008430588383934769, −4.05309356790400427773761775055, −2.559337276289280624852747316807, −1.2342167443628127356435283733, 2.990353165296785684561394969682, 4.038452369961345511602125612591, 4.89138392843139088667582573638, 6.48778067724624486046859183711, 7.94496714291411682300350918174, 8.529309934006062954206060022787, 10.56891514200976953003002653716, 11.26530519382185382142084345210, 12.81324055432860022236043909302, 14.165227674436104008425416390337, 14.94422087743868444182768674112, 15.61300129841492519660853981630, 16.74724165227280390930439452090, 17.71926236350683969002734879630, 19.422775062354228354968159686714, 20.58566435492126963035226049537, 21.33950193724705163117113599377, 22.539529748205529315485833153663, 23.24318983810889199092636967605, 24.20540290321050755132332288804, 25.58709658058898079727622287625, 26.34136537717902979328707329386, 27.217986275946530045730834065313, 27.9274189992557890495103395057, 29.94493471505290620259341206687

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