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Properties

Label 1-967-967.87-r0-0-0
Degree $1$
Conductor $967$
Sign $0.875 - 0.483i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.750 + 0.660i)2-s + (−0.638 − 0.769i)3-s + (0.126 − 0.991i)4-s + (0.981 − 0.193i)5-s + (0.987 + 0.155i)6-s + (0.874 − 0.485i)7-s + (0.560 + 0.828i)8-s + (−0.184 + 0.982i)9-s + (−0.608 + 0.793i)10-s + (−0.883 + 0.468i)11-s + (−0.844 + 0.536i)12-s + (−0.883 − 0.468i)13-s + (−0.334 + 0.942i)14-s + (−0.775 − 0.631i)15-s + (−0.967 − 0.250i)16-s + (−0.977 − 0.212i)17-s + ⋯
L(s)  = 1  + (−0.750 + 0.660i)2-s + (−0.638 − 0.769i)3-s + (0.126 − 0.991i)4-s + (0.981 − 0.193i)5-s + (0.987 + 0.155i)6-s + (0.874 − 0.485i)7-s + (0.560 + 0.828i)8-s + (−0.184 + 0.982i)9-s + (−0.608 + 0.793i)10-s + (−0.883 + 0.468i)11-s + (−0.844 + 0.536i)12-s + (−0.883 − 0.468i)13-s + (−0.334 + 0.942i)14-s + (−0.775 − 0.631i)15-s + (−0.967 − 0.250i)16-s + (−0.977 − 0.212i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.875 - 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9193911785 - 0.2368531377i\)
\(L(\frac12)\) \(\approx\) \(0.9193911785 - 0.2368531377i\)
\(L(1)\) \(\approx\) \(0.7525792964 - 0.04853888538i\)
\(L(1)\) \(\approx\) \(0.7525792964 - 0.04853888538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.750 + 0.660i)T \)
3 \( 1 + (-0.638 - 0.769i)T \)
5 \( 1 + (0.981 - 0.193i)T \)
7 \( 1 + (0.874 - 0.485i)T \)
11 \( 1 + (-0.883 + 0.468i)T \)
13 \( 1 + (-0.883 - 0.468i)T \)
17 \( 1 + (-0.977 - 0.212i)T \)
19 \( 1 + (0.241 + 0.970i)T \)
23 \( 1 + (0.938 + 0.344i)T \)
29 \( 1 + (-0.260 + 0.965i)T \)
31 \( 1 + (0.996 + 0.0779i)T \)
37 \( 1 + (0.999 + 0.0390i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (0.00975 - 0.999i)T \)
47 \( 1 + (0.425 + 0.905i)T \)
53 \( 1 + (0.854 - 0.519i)T \)
59 \( 1 + (-0.995 - 0.0974i)T \)
61 \( 1 + (-0.0682 - 0.997i)T \)
67 \( 1 + (0.996 - 0.0779i)T \)
71 \( 1 + (-0.750 - 0.660i)T \)
73 \( 1 + (0.854 + 0.519i)T \)
79 \( 1 + (0.316 - 0.948i)T \)
83 \( 1 + (0.737 - 0.675i)T \)
89 \( 1 + (0.996 - 0.0779i)T \)
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

−21.64741961082930896464728653870, −21.183280383725511531419615617168, −20.48556968134050543847099038759, −19.43792440862719720485527840318, −18.3326245463407654409805408165, −17.971500668240293491545886936634, −17.16164778850873156071245457611, −16.70076357411322633154133168566, −15.52870599354801148604464028096, −14.92447628024060949664957824382, −13.666172069548894032394272367260, −12.90993674840906241429331579172, −11.79927177657390796647671737610, −11.173424918803524497881754189868, −10.59852011336935585713733057198, −9.6862934830977544328300916316, −9.117482334921547913608128682193, −8.251639232044971199816093808491, −7.03914729623717728642860697708, −6.08757920101069389827004038966, −4.98107346734312570683485829756, −4.43088743788837875707896608495, −2.790630218326881400382723430328, −2.35179722645510203899853364284, −0.923632840773515191165090907484, 0.747187602423825858124828555, 1.76947246164810824416238227314, 2.44762823707946601282762811850, 4.86472424310384887793000452947, 5.12617397740416111871472121886, 6.0789810332625584852497254564, 7.07024623207575491460155562671, 7.60427085170305719454528053118, 8.45059305449093764366728508622, 9.52315857113119957458738828642, 10.49190676855067988403399925709, 10.84254389559401902360462428639, 12.061034569281693209204070534557, 13.045024164797552589172184425967, 13.77213949723346332589540741590, 14.51979116633873030643233019382, 15.46105735049441820833444506424, 16.52842129363687547214318480603, 17.25114581706458447130818177787, 17.63053943791064283097263633850, 18.23873050067958592505392695281, 18.968750361683067860007516189225, 20.12685238445145759031357424378, 20.62536919817430436775679163195, 21.77974554003553014978606036014

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