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Properties

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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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

\(L(\frac{1}{2})\) \(\approx\) \(0.08239058748 - 0.4832057699i\)
\(L(\frac12)\) \(\approx\) \(0.08239058748 - 0.4832057699i\)
\(L(1)\) \(\approx\) \(0.4354791097 - 0.1795350359i\)
\(L(1)\) \(\approx\) \(0.4354791097 - 0.1795350359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.961 + 0.276i)T \)
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 \( 1 + (0.260 - 0.965i)T \)
29 \( 1 + (-0.981 + 0.193i)T \)
31 \( 1 + (0.549 - 0.835i)T \)
37 \( 1 + (0.851 + 0.525i)T \)
41 \( 1 + (0.334 - 0.942i)T \)
43 \( 1 + (0.992 - 0.123i)T \)
47 \( 1 + (0.197 + 0.980i)T \)
53 \( 1 + (0.803 + 0.595i)T \)
59 \( 1 + (-0.982 + 0.187i)T \)
61 \( 1 + (-0.648 - 0.761i)T \)
67 \( 1 + (0.998 - 0.0585i)T \)
71 \( 1 + (0.241 + 0.970i)T \)
73 \( 1 + (0.803 - 0.595i)T \)
79 \( 1 + (-0.401 - 0.915i)T \)
83 \( 1 + (0.471 - 0.881i)T \)
89 \( 1 + (-0.448 + 0.893i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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