L-function 1-475-475.446-r0-0-0

• Label: 1-475-475.446-r0-0-0
• Degree: $1$
• Conductor: $475$
• Sign: $0.998 + 0.0576i$
• Analytic cond.: $2.20589$
• Root an. cond.: $2.20589$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.961 + 0.275i)2^-s + (0.990 − 0.139i)3^-s + (0.848 + 0.529i)4^-s + (0.990 + 0.139i)6^-s + (−0.5 − 0.866i)7^-s + (0.669 + 0.743i)8^-s + (0.961 − 0.275i)9^-s + (0.913 − 0.406i)11^-s + (0.913 + 0.406i)12^-s + (−0.719 − 0.694i)13^-s + (−0.241 − 0.970i)14^-s + (0.438 + 0.898i)16^-s + (0.0348 + 0.999i)17^-s + 18^-s + (−0.615 − 0.788i)21^-s + (0.990 − 0.139i)22^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0576i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$475$    =    $5^{2} \cdot 19$
Sign:	$0.998 + 0.0576i$
Analytic conductor:	$2.20589$
Root analytic conductor:	$2.20589$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{475} (446, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 475,\ (0:\ ),\ 0.998 + 0.0576i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$3.367907878 + 0.09718019789i$
$L(1)$	$\approx$	$2.423982070 + 0.1236179477i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	19	$1$
good	2	$1 + (0.961 + 0.275i)T$
	3	$1 + (0.990 - 0.139i)T$
	7	$1 + (-0.5 - 0.866i)T$
	11	$1 + (0.913 - 0.406i)T$
	13	$1 + (-0.719 - 0.694i)T$
	17	$1 + (0.0348 + 0.999i)T$
	23	$1 + (-0.997 + 0.0697i)T$
	29	$1 + (0.0348 - 0.999i)T$
	31	$1 + (-0.978 + 0.207i)T$

Imaginary part of the first few zeros on the critical line

−24.072790768017155160653570555319, −22.56813888188848796756936701263, −22.162557501041487601880119980623, −21.33266044600726083632688194464, −20.467024725549314518736724849663, −19.65899769897401996476241960717, −19.14228455710943111759609106748, −18.11000384245336084120078798744, −16.46194028430365155921139623647, −15.86798596967674993642697420560, −14.86530881900598179961201068671, −14.3321833545809605569110432489, −13.55041649502436602273816222469, −12.32595406568481984816101954551, −12.0629424966966436753193447359, −10.64086470145844778608638420694, −9.4839664163828505296362569742, −9.0768137344225304680762542372, −7.45107742161681590155716081404, −6.76118891765643591065910654603, −5.5212339156288306461150601355, −4.43394338303250573806082845471, −3.562057463010790398162709642086, −2.5170306258488624231223449743, −1.77681216165633993428956645270, 1.50477190303074012426238605876, 2.76725176783653293937763245474, 3.7299077290201284913895256069, 4.28686513003252828572843727054, 5.83114400494392154013405186962, 6.77097013528425622030547084618, 7.60966341565346328497411056127, 8.42683725809734121984028259599, 9.74657805982270889787755448502, 10.59970476439943450557553051965, 11.91859784322842887678771674828, 12.80483397698279214428580069937, 13.48575995472114246912154452379, 14.30879019289030139803659086958, 14.90742927671938863629407898930, 15.87680645762233929094289789314, 16.775045955611755651720794007817, 17.60694958336447261148910348045, 19.16678642423014145261233126613, 19.83744202641803317125046026080, 20.29219455710117840274712068688, 21.42580201728133360579735970463, 22.09507863411967904053578647028, 22.99974974447234591900760558972, 23.97949333901658133757340537330

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