L-function 1-320-320.3-r0-0-0

• Label: 1-320-320.3-r0-0-0
• Degree: $1$
• Conductor: $320$
• Sign: $-0.502 + 0.864i$
• Analytic cond.: $1.48607$
• Root an. cond.: $1.48607$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.382 + 0.923i)3^-s + (0.707 + 0.707i)7^-s + (−0.707 − 0.707i)9^-s + (−0.923 + 0.382i)11^-s + (0.923 + 0.382i)13^-s + 17^-s + (−0.382 + 0.923i)19^-s + (−0.923 + 0.382i)21^-s + (−0.707 + 0.707i)23^-s + (0.923 − 0.382i)27^-s + (−0.923 − 0.382i)29^-s + 31^-s − i·33^-s + (−0.923 + 0.382i)37^-s + (−0.707 + 0.707i)39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(320\)    =    \(2^{6} \cdot 5\)
Sign:	$-0.502 + 0.864i$
Analytic conductor:	\(1.48607\)
Root analytic conductor:	\(1.48607\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{320} (3, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 320,\ (0:\ ),\ -0.502 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.5017637674 + 0.8718459638i\)
\(L(1)\)	\(\approx\)	\(0.8183212376 + 0.4558308855i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	5	\( 1 \)
good	3	\( 1 + (-0.382 + 0.923i)T \)
	7	\( 1 + (0.707 + 0.707i)T \)
	11	\( 1 + (-0.923 + 0.382i)T \)
	13	\( 1 + (0.923 + 0.382i)T \)
	17	\( 1 + T \)
	19	\( 1 + (-0.382 + 0.923i)T \)
	23	\( 1 + (-0.707 + 0.707i)T \)
	29	\( 1 + (-0.923 - 0.382i)T \)
	31	\( 1 + T \)

Imaginary part of the first few zeros on the critical line

−24.62777805748939081360727916437, −23.82661104996601435362161290760, −23.36250278143740542657827439800, −22.39718480721994878752799960597, −21.116604156419735651029747844267, −20.438558299741642466502568434247, −19.31930918075323143014438492362, −18.42278365016264736531234103980, −17.78409795740037628185888057766, −16.85744673321071215234280592306, −15.95456716983012432862895129290, −14.63599497398326865167763392894, −13.64815709571221561579153825111, −13.077844192981583710738391407386, −11.9264443028240875024330698159, −10.97483655258909927462974250336, −10.32388893344471970824466944979, −8.50680537398287716754885666531, −7.91141588379183710342812399863, −6.90666929038235930111112736368, −5.79062285794098302407730690306, −4.85939107983190506330707322333, −3.32844979881181375301103282368, −1.94937688840660461366794030940, −0.692863331025744953383754183483, 1.72079154634831705871647142900, 3.21184535919321349899278745906, 4.339686313348678559563642837440, 5.40612260192459299207251559104, 6.0698667692528441506956291992, 7.783950752608875011624614164084, 8.630134606606667756240244818808, 9.77201492767842929637269624542, 10.57402587239495451553266138305, 11.58861542677078500663138802599, 12.297996795088241375369002660591, 13.718267716689510384095376821286, 14.74886966479028896309531263795, 15.5063184359915760583298226312, 16.269594987862398948096006622165, 17.29551533283879001571405354134, 18.20230637331198147140469173908, 18.98335300910701694968903262099, 20.569463592607458620489676728999, 21.00133661504798199635280615767, 21.66800114100650602172885787883, 22.87128779366435981667235797088, 23.42422164184174521182491251048, 24.497095064555669718922258897991, 25.73688411858253795024732727343

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