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Properties

Label 1-11-11.2-r1-0-0
Degree $1$
Conductor $11$
Sign $0.957 + 0.288i$
Analytic cond. $1.18211$
Root an. cond. $1.18211$
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  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + 12-s + (0.809 + 0.587i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.809 − 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s − 10-s + 12-s + (0.809 + 0.587i)13-s + (0.309 − 0.951i)14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(11\)
Sign: $0.957 + 0.288i$
Analytic conductor: \(1.18211\)
Root analytic conductor: \(1.18211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{11} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 11,\ (1:\ ),\ 0.957 + 0.288i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.503380081 + 0.2211997697i\)
\(L(\frac12)\) \(\approx\) \(1.503380081 + 0.2211997697i\)
\(L(1)\) \(\approx\) \(1.415747663 + 0.1740328837i\)
\(L(1)\) \(\approx\) \(1.415747663 + 0.1740328837i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−45.30426327898638018804634367866, −43.63192017509203526976799880310, −42.45917275651500585448292842681, −40.66058066592356276517670187998, −39.22705738819166212922797709288, −38.395494452428109122979398955656, −37.01755954390176887573149124687, −34.72345992628301281730177232071, −32.799985112149809068419874207202, −31.87403308116394856064712507976, −30.758854343797428801601385292783, −28.4508199966808353191563583286, −27.61231692090569895478179228700, −25.294856636069449525514142678791, −23.38804121013922687721723923210, −21.88489757248339392930656813056, −20.5790557747237967713122828270, −19.23890660918179306832270433212, −16.003353712811180942448748166867, −14.94827047321579871059739778452, −12.73163403755865549814505940274, −11.00919009252031218454364175306, −8.95354546437232036116473143789, −5.27308657584208989337435824912, −3.41492187932220368062022478551, 3.54704109171945007666447637176, 6.63073045048494212311693469779, 7.88643465920150753487622877767, 11.59406426254877199887647483811, 13.33022743788218395645836666989, 14.674032161018154187712356107968, 16.64292363517938135933695505995, 18.7495375665351457827169284855, 20.50797618286869986017769960585, 22.95929393378876961523787194156, 23.62369920986579610417564305629, 25.38976369113433935050441838210, 26.67028496809013083238174485224, 29.55920151127941976080946946763, 30.615315398535949332903421061066, 31.76704292714201836153420720265, 33.60918441512861464269747264056, 35.0748181390130967490481952242, 36.103221852765725499410258250258, 38.421569250426374648790601150719, 39.91816544484692915524132036871, 41.34346950295061398221815956035, 42.601218189354830328691114239797, 43.13903733098448016886778132619, 45.571883708328470662009967870960

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