L-function 1-967-967.63-r1-0-0

• Label: 1-967-967.63-r1-0-0
• Degree: $1$
• Conductor: $967$
• Sign: $-0.943 - 0.331i$
• Analytic cond.: $103.918$
• Root an. cond.: $103.918$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.961 + 0.276i)2^-s + (−0.126 − 0.991i)3^-s + (0.847 − 0.530i)4^-s + (−0.929 − 0.368i)5^-s + (0.395 + 0.918i)6^-s + (−0.989 − 0.142i)7^-s + (−0.668 + 0.744i)8^-s + (−0.967 + 0.250i)9^-s + (0.995 + 0.0974i)10^-s + (−0.407 + 0.913i)11^-s + (−0.633 − 0.773i)12^-s + (−0.994 − 0.103i)13^-s + (0.990 − 0.136i)14^-s + (−0.247 + 0.968i)15^-s + (0.436 − 0.899i)16^-s + (0.811 − 0.584i)17^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$967$
Sign:	$-0.943 - 0.331i$
Analytic conductor:	$103.918$
Root analytic conductor:	$103.918$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{967} (63, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 967,\ (1:\ ),\ -0.943 - 0.331i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.08239058748 - 0.4832057699i$
$L(1)$	$\approx$	$0.4354791097 - 0.1795350359i$

Euler product

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

	$p$	$F_p(T)$
bad	967	$1$
good	2	$1 + (-0.961 + 0.276i)T$
	3	$1 + (-0.126 - 0.991i)T$
	5	$1 + (-0.929 - 0.368i)T$
	7	$1 + (-0.989 - 0.142i)T$
	11	$1 + (-0.407 + 0.913i)T$
	13	$1 + (-0.994 - 0.103i)T$
	17	$1 + (0.811 - 0.584i)T$
	19	$1 + (0.454 - 0.890i)T$
	23	$1 + (0.260 - 0.965i)T$

Imaginary part of the first few zeros on the critical line

−21.62381037723960165273399795466, −21.23779131373823019122792199093, −20.03767114336389910144927763533, −19.546495579804327496522183548778, −18.94948115370584472044950239510, −18.11281217768680914991907688824, −16.72328856960556151737918091612, −16.64190953959093179814315312243, −15.715732715400900829656613114991, −15.1755230369202913967623330792, −14.22689002722048873618304098126, −12.7610596224764195873082471655, −11.98183386580388206376879428042, −11.293903789898949549425411010096, −10.456995547925287206688572465015, −9.84069796107150823493058770200, −9.10055922465137982559024901892, −8.095168481880896112568039128387, −7.449401778003126318533572052933, −6.273032610074849809899190007563, −5.43544635935711226753177716003, −3.872750726450776811131998853566, −3.357782837807105221044933732218, −2.599535993845541071706104399029, −0.75046746610369619253366140159, 0.2806108068075089548449796541, 0.84067857997581021804583527239, 2.34599432783112172849312370795, 3.012061937734818113442756218417, 4.66825690810097333929965643137, 5.673464441590341622527839472375, 6.75099885381924575383700720551, 7.4978443747534637064684532996, 7.7016313870951158444301396891, 9.00273361817021837145227511416, 9.62215639769516406020899209570, 10.68172962666924532650263319064, 11.632119115476066225880813005494, 12.37214432967680462031083067036, 12.86062722212216360388156739366, 14.14324075172994096547255545478, 15.114965741946433626250432839589, 15.78208713548804716966458395859, 16.769174847955117312571765453098, 17.11674744941126618885943493433, 18.24059659534480739468332166958, 18.84956862543106818025443344360, 19.44581197941489838669789956878, 20.18773832241794071405082058921, 20.581358632323252085627621389291

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