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Properties

Label 1-967-967.167-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.771 + 0.636i$
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  + (0.126 − 0.991i)2-s + (0.184 − 0.982i)3-s + (−0.967 − 0.250i)4-s + (−0.924 + 0.380i)5-s + (−0.951 − 0.307i)6-s + (−0.527 + 0.849i)7-s + (−0.371 + 0.928i)8-s + (−0.932 − 0.362i)9-s + (0.260 + 0.965i)10-s + (0.560 − 0.828i)11-s + (−0.425 + 0.905i)12-s + (−0.560 − 0.828i)13-s + (0.775 + 0.631i)14-s + (0.203 + 0.979i)15-s + (0.874 + 0.485i)16-s + (0.909 + 0.416i)17-s + ⋯
L(s)  = 1  + (0.126 − 0.991i)2-s + (0.184 − 0.982i)3-s + (−0.967 − 0.250i)4-s + (−0.924 + 0.380i)5-s + (−0.951 − 0.307i)6-s + (−0.527 + 0.849i)7-s + (−0.371 + 0.928i)8-s + (−0.932 − 0.362i)9-s + (0.260 + 0.965i)10-s + (0.560 − 0.828i)11-s + (−0.425 + 0.905i)12-s + (−0.560 − 0.828i)13-s + (0.775 + 0.631i)14-s + (0.203 + 0.979i)15-s + (0.874 + 0.485i)16-s + (0.909 + 0.416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.771 + 0.636i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.771 + 0.636i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3674182187 - 1.022531490i\)
\(L(\frac12)\) \(\approx\) \(-0.3674182187 - 1.022531490i\)
\(L(1)\) \(\approx\) \(0.5624431204 - 0.6403204808i\)
\(L(1)\) \(\approx\) \(0.5624431204 - 0.6403204808i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.126 - 0.991i)T \)
3 \( 1 + (0.184 - 0.982i)T \)
5 \( 1 + (-0.924 + 0.380i)T \)
7 \( 1 + (-0.527 + 0.849i)T \)
11 \( 1 + (0.560 - 0.828i)T \)
13 \( 1 + (-0.560 - 0.828i)T \)
17 \( 1 + (0.909 + 0.416i)T \)
19 \( 1 + (0.883 - 0.468i)T \)
23 \( 1 + (-0.763 - 0.646i)T \)
29 \( 1 + (0.864 + 0.502i)T \)
31 \( 1 + (0.987 + 0.155i)T \)
37 \( 1 + (-0.996 - 0.0779i)T \)
41 \( 1 + (0.990 - 0.136i)T \)
43 \( 1 + (0.999 + 0.0195i)T \)
47 \( 1 + (0.638 - 0.769i)T \)
53 \( 1 + (0.460 - 0.887i)T \)
59 \( 1 + (0.981 + 0.193i)T \)
61 \( 1 + (-0.990 + 0.136i)T \)
67 \( 1 + (-0.987 + 0.155i)T \)
71 \( 1 + (0.126 + 0.991i)T \)
73 \( 1 + (0.460 + 0.887i)T \)
79 \( 1 + (0.799 + 0.600i)T \)
83 \( 1 + (0.0876 - 0.996i)T \)
89 \( 1 + (-0.987 + 0.155i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−22.51882640200967433211301410296, −21.23964314558466461529870392028, −20.54805067890945761108140791510, −19.57343729286402048412864664552, −19.15977193784753519742801959604, −17.69608524013392257652493600675, −16.97563191326634861761430099028, −16.332029128074023275503972799368, −15.848131683186794681583845750808, −15.07450481032197340698927580543, −14.12146196145889984006525399634, −13.79932775077861478699638122331, −12.246010286896186444709964061052, −11.92475256430657168798844709761, −10.44812017351037555378026310934, −9.58058750710602587183339892842, −9.21262884051238855288319338213, −7.86662174982128610561092593759, −7.50724319482419240871785585119, −6.431873092043752305407005445894, −5.28281708793795117273580319676, −4.34805358679316978994022068949, −4.02502443658290632234030924440, −3.04981343335432657585082460608, −0.949891019377667089443947378434, 0.3120493965389056337588874500, 1.07377698776684082321319451149, 2.57190206040270202350420566947, 3.01151944177328491071449103785, 3.861366380325492896213707540284, 5.31501979379698169067656684938, 6.072108805399785732282723275108, 7.17145202300943839563120125016, 8.25954063401638830065183919480, 8.65798921985370877853271922529, 9.81821569319243820122719437837, 10.767702768800010563456009122440, 11.79123981383768270756013401488, 12.156373907158729601945141686849, 12.72541266596194621701388508715, 13.86879141950548609359642709798, 14.42055841962788564391376101190, 15.315559697601386199167082833239, 16.348788819678954424351055037917, 17.56558143827236715042804214225, 18.23610564532100573695554005619, 18.99845956602733385296799942519, 19.469934117062983014363750035906, 19.910213412900298827246274881217, 20.97367749649396215188994821925

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