L-function 1-399-399.257-r1-0-0

• Label: 1-399-399.257-r1-0-0
• Degree: $1$
• Conductor: $399$
• Sign: $-0.243 + 0.970i$
• Analytic cond.: $42.8785$
• Root an. cond.: $42.8785$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.766 + 0.642i)2^-s + (0.173 + 0.984i)4^-s + (0.173 − 0.984i)5^-s + (−0.5 + 0.866i)8^-s + (0.766 − 0.642i)10^-s + (0.5 + 0.866i)11^-s + (0.173 + 0.984i)13^-s + (−0.939 + 0.342i)16^-s + (0.173 − 0.984i)17^-s + 20^-s + (−0.173 + 0.984i)22^-s + (0.939 + 0.342i)23^-s + (−0.939 − 0.342i)25^-s + (−0.5 + 0.866i)26^-s + (−0.939 − 0.342i)29^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.243 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$399$    =    $3 \cdot 7 \cdot 19$
Sign:	$-0.243 + 0.970i$
Analytic conductor:	$42.8785$
Root analytic conductor:	$42.8785$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{399} (257, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 399,\ (1:\ ),\ -0.243 + 0.970i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.897083969 + 2.431003502i$
$L(1)$	$\approx$	$1.523771768 + 0.7623433813i$

Euler product

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

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
	19	$1$
good	2	$1 + (0.766 + 0.642i)T$
	5	$1 + (0.173 - 0.984i)T$
	11	$1 + (0.5 + 0.866i)T$
	13	$1 + (0.173 + 0.984i)T$
	17	$1 + (0.173 - 0.984i)T$
	23	$1 + (0.939 + 0.342i)T$
	29	$1 + (-0.939 - 0.342i)T$
	31	$1 + T$
	37	$1 + (0.5 + 0.866i)T$

Imaginary part of the first few zeros on the critical line

−23.71527180853680704806838223344, −22.79409698250779078948194970715, −22.23508842321731306760680555771, −21.44983618941702005693111444274, −20.62921379006470391141749360406, −19.47165463060949527895060469391, −18.98583736256996458368846652097, −18.03767769590822718894516314974, −16.94220448629933485968710200602, −15.59192976069930230131732887, −14.88814434067470300438822327657, −14.12555630955056254833764701957, −13.25519957898194290095700625257, −12.363121381031987326708291395852, −11.15569471632545097221834810923, −10.73422059743056903853802659609, −9.77204238578841687598981185910, −8.52095425432212813081457025606, −7.10395644031210534093585764263, −6.12703117486389830469345511472, −5.409622930963156037611426732542, −3.85453822580741799813794570058, −3.2076485881996041504086179373, −2.091129705390256824631621715035, −0.6768337846858851872849963486, 1.32914830768348015437254836934, 2.716785432608466913516503172344, 4.2264162052675079701789187806, 4.73468314071697428025628908411, 5.851352382107150107867272812834, 6.88483546464292399774312392835, 7.79416530889055543237399059160, 8.978089689672732559683488878657, 9.603353553407017351742405004064, 11.43581287289433006653224616369, 12.03593861800740249235364535662, 13.023521340957214902288773787567, 13.72195041943808274681363434444, 14.68404094139928320795900677374, 15.61020650307502643702110224816, 16.523811810266133349789037455903, 17.080392996730397112039702590437, 18.00078769221188442121111692504, 19.32093503824038086305914149812, 20.51888837618255052212519163939, 20.90992369271579633988758409935, 21.91078113727774383475224398637, 22.86732739311919317032264888149, 23.55800282062283644044265609924, 24.483323484206518221554707220320

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