L-function 1-4235-4235.1077-r1-0-0

• Label: 1-4235-4235.1077-r1-0-0
• Degree: $1$
• Conductor: $4235$
• Sign: $-0.981 - 0.188i$
• Analytic cond.: $455.113$
• Root an. cond.: $455.113$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.909 − 0.415i)2^-s + i·3^-s + (0.654 + 0.755i)4^-s + (0.415 − 0.909i)6^-s + (−0.281 − 0.959i)8^-s − 9^-s + (−0.755 + 0.654i)12^-s + (−0.755 + 0.654i)13^-s + (−0.142 + 0.989i)16^-s + (−0.540 + 0.841i)17^-s + (0.909 + 0.415i)18^-s + (−0.841 + 0.540i)19^-s + (0.989 + 0.142i)23^-s + (0.959 − 0.281i)24^-s + (0.959 − 0.281i)26^-s − i·27^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign:	$-0.981 - 0.188i$
Analytic conductor:	\(455.113\)
Root analytic conductor:	\(455.113\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4235} (1077, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 4235,\ (1:\ ),\ -0.981 - 0.188i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(-0.04865174865 + 0.5103542989i\)
\(L(1)\)	\(\approx\)	\(0.5914656042 + 0.1812543245i\)

Euler product

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

	$p$	$F_p(T)$
bad	5	\( 1 \)
	7	\( 1 \)
	11	\( 1 \)
good	2	\( 1 + (-0.909 - 0.415i)T \)
	3	\( 1 + iT \)
	13	\( 1 + (-0.755 + 0.654i)T \)
	17	\( 1 + (-0.540 + 0.841i)T \)
	19	\( 1 + (-0.841 + 0.540i)T \)
	23	\( 1 + (0.989 + 0.142i)T \)
	29	\( 1 + (0.841 - 0.540i)T \)
	31	\( 1 + (0.654 - 0.755i)T \)
	37	\( 1 + (-0.755 - 0.654i)T \)

Imaginary part of the first few zeros on the critical line

−17.73699444847475403555936742075, −17.3608348280371312380550640008, −16.801789338480301715768712857570, −15.79803943123762000557160772272, −15.31180724223583955073531418068, −14.38852707968852200053010410162, −14.00492051752525402155915844516, −12.91112992629314739626062688504, −12.49625761476141407231294050109, −11.47869595815496197104427518451, −11.07356984677392398094790382544, −10.20039318498236432024573259205, −9.417281954436751259215834662051, −8.644952547778229644006929869314, −8.16008039918249872246806231801, −7.36719584948212012190089784786, −6.7112303479032275105694854424, −6.371013576333824110566040620471, −5.181340075322143208637420904738, −4.80694170335741029548186768266, −2.98606598288568539227650859880, −2.65174988350110697454109758021, −1.65961333063843263945302159563, −0.80984049470867433389081407639, −0.148693263802345838393316771569, 0.83691149566344475024617205679, 2.14708880139962282191321017084, 2.51497107618738234738664558731, 3.72484449297744018976245199234, 4.10336028338214955621212429093, 5.0436919551006002628158115060, 6.058530492250754887554003834943, 6.75760685012184088093938950703, 7.6329358439380597211226247357, 8.53088627165693672761504399109, 8.911399162682597680323188370257, 9.65614659011676580298041006682, 10.38609978446050765829092961124, 10.72995210381387947193393946162, 11.59886689678355929605125779406, 12.147669605398161316707219227241, 12.93745636589099913175995591994, 13.90235824101465576178556014613, 14.72861943851180076166122462487, 15.40252901702001107174452243793, 15.86283132492679234341930575411, 16.851708945533469375518292220681, 17.134429143230663612351797104323, 17.60058221254008836348876194356, 18.8056603036524964240794846128

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