L-function 1-595-595.123-r1-0-0

• Label: 1-595-595.123-r1-0-0
• Degree: $1$
• Conductor: $595$
• Sign: $-0.539 + 0.842i$
• Analytic cond.: $63.9416$
• Root an. cond.: $63.9416$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.866 − 0.5i)2^-s + (−0.5 − 0.866i)3^-s + (0.5 + 0.866i)4^-s + i·6^-s − i·8^-s + (−0.5 + 0.866i)9^-s + (0.866 − 0.5i)11^-s + (0.5 − 0.866i)12^-s + i·13^-s + (−0.5 + 0.866i)16^-s + (0.866 − 0.5i)18^-s + (−0.5 + 0.866i)19^-s − 22^-s + (0.5 − 0.866i)23^-s + (−0.866 + 0.5i)24^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(595\)    =    \(5 \cdot 7 \cdot 17\)
Sign:	$-0.539 + 0.842i$
Analytic conductor:	\(63.9416\)
Root analytic conductor:	\(63.9416\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{595} (123, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 595,\ (1:\ ),\ -0.539 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.01065811558 + 0.01948087263i\)
\(L(1)\)	\(\approx\)	\(0.5156386214 - 0.1995963013i\)

Euler product

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

	$p$	$F_p(T)$
bad	5	\( 1 \)
	7	\( 1 \)
	17	\( 1 \)
good	2	\( 1 + (-0.866 - 0.5i)T \)
	3	\( 1 + (-0.5 - 0.866i)T \)
	11	\( 1 + (0.866 - 0.5i)T \)
	13	\( 1 + iT \)
	19	\( 1 + (-0.5 + 0.866i)T \)
	23	\( 1 + (0.5 - 0.866i)T \)
	29	\( 1 + iT \)
	31	\( 1 + (-0.866 + 0.5i)T \)
	37	\( 1 + (0.5 - 0.866i)T \)

Imaginary part of the first few zeros on the critical line

−22.724394183523436350872337957327, −21.899250065290876681279370464621, −20.81810623625342823440768433626, −20.082881504346582484318814088367, −19.36948415022370901337710461412, −18.18549899091617253859602077466, −17.34730916343335886231362379607, −17.04496402689820395829074246695, −15.96962567486768722532163097395, −15.156823217585509344508539214574, −14.82051345619748972983581650912, −13.46149971371436691674203914430, −12.1016890003355366225973844470, −11.24441824112440298022135618236, −10.53242116129024799518076218448, −9.58044903469910735850174030417, −9.07682854422714421948919735356, −7.93208309140033742072836041330, −6.88140831177430704500793878380, −5.98267205439075899357079516516, −5.13952560925442208903667530608, −4.07854313543728200719015461825, −2.70461784100553788333494249880, −1.16939582693500338396513687717, −0.008875509353769136659016023175, 1.23907440037101337143091558225, 1.972003956373525735055178151617, 3.24549857846826420937806482913, 4.45963244846481998505420375166, 6.0314637211523011271523750772, 6.73390308972555393383767887872, 7.57917061541662367492707967223, 8.629067046428876180132983878231, 9.262259438546583734826180567490, 10.656002477167397858885712525405, 11.15464198150208807514503397070, 12.171594452578660484274506691014, 12.600490389268104055466632282536, 13.79011840462115198665281446503, 14.6512365670209106556840356470, 16.321926239677502692140161397280, 16.62308979558569177805344478329, 17.44716552022387634347257830337, 18.425179361427753598297945874408, 18.91562446682281873067562612688, 19.630931853655022325964718539450, 20.49264559204764067103496970434, 21.61754466172957086286028949262, 22.16410802501220523145348685646, 23.27516687134391674753355579982

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