L-function 1-11-11.2-r1-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	11	$1$
good	2	$1 + (0.809 + 0.587i)T$
	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	$1 + (-0.309 - 0.951i)T$

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