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Properties

Label 1-967-967.199-r0-0-0
Degree $1$
Conductor $967$
Sign $0.437 - 0.899i$
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  + (0.581 + 0.813i)2-s + (−0.260 − 0.965i)3-s + (−0.322 + 0.946i)4-s + (0.549 − 0.835i)5-s + (0.633 − 0.773i)6-s + (−0.964 + 0.263i)7-s + (−0.957 + 0.288i)8-s + (−0.864 + 0.502i)9-s + (0.999 − 0.0390i)10-s + (−0.145 + 0.989i)11-s + (0.997 + 0.0649i)12-s + (0.929 + 0.368i)13-s + (−0.775 − 0.631i)14-s + (−0.949 − 0.313i)15-s + (−0.791 − 0.610i)16-s + (−0.638 − 0.769i)17-s + ⋯
L(s)  = 1  + (0.581 + 0.813i)2-s + (−0.260 − 0.965i)3-s + (−0.322 + 0.946i)4-s + (0.549 − 0.835i)5-s + (0.633 − 0.773i)6-s + (−0.964 + 0.263i)7-s + (−0.957 + 0.288i)8-s + (−0.864 + 0.502i)9-s + (0.999 − 0.0390i)10-s + (−0.145 + 0.989i)11-s + (0.997 + 0.0649i)12-s + (0.929 + 0.368i)13-s + (−0.775 − 0.631i)14-s + (−0.949 − 0.313i)15-s + (−0.791 − 0.610i)16-s + (−0.638 − 0.769i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.437 - 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.054619206 - 0.6597296450i\)
\(L(\frac12)\) \(\approx\) \(1.054619206 - 0.6597296450i\)
\(L(1)\) \(\approx\) \(1.105487467 + 0.01584826853i\)
\(L(1)\) \(\approx\) \(1.105487467 + 0.01584826853i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.581 + 0.813i)T \)
3 \( 1 + (-0.260 - 0.965i)T \)
5 \( 1 + (0.549 - 0.835i)T \)
7 \( 1 + (-0.964 + 0.263i)T \)
11 \( 1 + (-0.145 + 0.989i)T \)
13 \( 1 + (0.929 + 0.368i)T \)
17 \( 1 + (-0.638 - 0.769i)T \)
19 \( 1 + (0.328 - 0.944i)T \)
23 \( 1 + (-0.407 - 0.913i)T \)
29 \( 1 + (0.996 + 0.0779i)T \)
31 \( 1 + (0.903 - 0.428i)T \)
37 \( 1 + (-0.677 - 0.735i)T \)
41 \( 1 + (-0.990 + 0.136i)T \)
43 \( 1 + (-0.0552 - 0.998i)T \)
47 \( 1 + (0.991 - 0.129i)T \)
53 \( 1 + (-0.998 + 0.0455i)T \)
59 \( 1 + (0.880 + 0.474i)T \)
61 \( 1 + (0.377 - 0.926i)T \)
67 \( 1 + (-0.822 + 0.568i)T \)
71 \( 1 + (-0.995 - 0.0974i)T \)
73 \( 1 + (-0.998 - 0.0455i)T \)
79 \( 1 + (0.266 - 0.963i)T \)
83 \( 1 + (0.999 + 0.0130i)T \)
89 \( 1 + (-0.0812 - 0.996i)T \)
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

−22.02162666924858727979821722712, −21.19577748472966328615376004623, −20.60950158014834234566018143729, −19.56501368209489101789156444633, −19.00694462392809442528813224614, −18.040134730886770595970512869264, −17.257950847940540227115872443808, −16.02281182000644441898897938823, −15.6060440736631045068527546010, −14.63599989737098142358934418111, −13.71110863294693980799390709994, −13.40586256559938632674891578612, −12.11421163486326739827608626626, −11.29228133275932570531372409509, −10.353714990789289652233867175379, −10.30616923033885559151462311611, −9.27083073269636366695872142112, −8.33369978284106415273618250989, −6.43882676319391050295854365163, −6.13072754344227681943827119578, −5.30456150828567131669237499105, −3.96342861471399200713821665887, −3.38712155161890642969883218368, −2.78742375579072630556754786302, −1.25882602731329392834248618987, 0.47183779635276481366233584339, 2.05835989967457260171801563240, 2.89650818588892262559746589152, 4.35567878160136953568976755220, 5.11565043650884930764099347606, 6.074348989932957914937052063742, 6.64627637966119998748095783534, 7.35638779172846208151921824969, 8.63485435438006521946649818654, 8.965659686989433760700742602193, 10.169374544229194601152930619033, 11.691800621442277226389236213013, 12.26282733517171681412741310107, 13.01620431856542677914235455311, 13.527439990196818810441348756321, 14.12800452530492655667448136695, 15.54140457222189158257764541273, 16.01436867046880537164310820872, 16.84990903347910836506082587861, 17.67557149309497362750381333770, 18.114538533290323065290577069619, 19.119242334993674445835439695607, 20.25565649964256461077845865630, 20.73228477079838105532225817959, 22.04867770442303409902979835173

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