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Properties

Label 1-967-967.80-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.967 - 0.253i$
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  + (−0.864 − 0.502i)2-s + (−0.165 + 0.986i)3-s + (0.494 + 0.869i)4-s + (−0.951 − 0.307i)5-s + (0.638 − 0.769i)6-s + (0.477 − 0.878i)7-s + (0.00975 − 0.999i)8-s + (−0.945 − 0.325i)9-s + (0.668 + 0.744i)10-s + (0.710 + 0.703i)11-s + (−0.938 + 0.344i)12-s + (−0.710 + 0.703i)13-s + (−0.854 + 0.519i)14-s + (0.460 − 0.887i)15-s + (−0.511 + 0.859i)16-s + (−0.0292 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.864 − 0.502i)2-s + (−0.165 + 0.986i)3-s + (0.494 + 0.869i)4-s + (−0.951 − 0.307i)5-s + (0.638 − 0.769i)6-s + (0.477 − 0.878i)7-s + (0.00975 − 0.999i)8-s + (−0.945 − 0.325i)9-s + (0.668 + 0.744i)10-s + (0.710 + 0.703i)11-s + (−0.938 + 0.344i)12-s + (−0.710 + 0.703i)13-s + (−0.854 + 0.519i)14-s + (0.460 − 0.887i)15-s + (−0.511 + 0.859i)16-s + (−0.0292 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.967 - 0.253i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.967 - 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01551671281 - 0.1205921067i\)
\(L(\frac12)\) \(\approx\) \(0.01551671281 - 0.1205921067i\)
\(L(1)\) \(\approx\) \(0.5369787056 + 0.01082390245i\)
\(L(1)\) \(\approx\) \(0.5369787056 + 0.01082390245i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.864 - 0.502i)T \)
3 \( 1 + (-0.165 + 0.986i)T \)
5 \( 1 + (-0.951 - 0.307i)T \)
7 \( 1 + (0.477 - 0.878i)T \)
11 \( 1 + (0.710 + 0.703i)T \)
13 \( 1 + (-0.710 + 0.703i)T \)
17 \( 1 + (-0.0292 - 0.999i)T \)
19 \( 1 + (-0.924 - 0.380i)T \)
23 \( 1 + (0.371 - 0.928i)T \)
29 \( 1 + (0.107 + 0.994i)T \)
31 \( 1 + (0.425 + 0.905i)T \)
37 \( 1 + (0.844 + 0.536i)T \)
41 \( 1 + (-0.203 - 0.979i)T \)
43 \( 1 + (0.799 - 0.600i)T \)
47 \( 1 + (-0.763 - 0.646i)T \)
53 \( 1 + (-0.0682 - 0.997i)T \)
59 \( 1 + (0.987 - 0.155i)T \)
61 \( 1 + (0.203 + 0.979i)T \)
67 \( 1 + (-0.425 + 0.905i)T \)
71 \( 1 + (-0.864 + 0.502i)T \)
73 \( 1 + (-0.0682 + 0.997i)T \)
79 \( 1 + (-0.737 - 0.675i)T \)
83 \( 1 + (0.241 - 0.970i)T \)
89 \( 1 + (-0.425 + 0.905i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−22.09974017260445981143548172432, −20.98602274633282504964362778387, −19.66410914547282570921319729707, −19.43330817212647059507439185625, −18.84086991803269926205737007161, −18.01262691963732727641890394469, −17.27260813318352948581565659506, −16.667973068455989708039962196318, −15.48824575524266820429578893334, −14.8936879420668833267208633150, −14.33649107950745883532927589255, −13.00189409084202700892994106773, −12.09231174730269655781551001532, −11.392527920899736712123014081418, −10.87623122310626709526467324783, −9.554889533602224312718878772919, −8.482843463208181871207915343215, −8.043986946422871514216667993488, −7.42016574982864914636384342455, −6.18161953930729877869955863956, −5.933676203785667940515764431589, −4.56852401521940577911847837021, −3.029529269540691264495312408931, −2.079694661722794159088307300779, −1.00238275061725591706501595195, 0.04799579385324920284360165012, 1.01108142641316316913015727766, 2.42150965546600886741639231145, 3.57311571809643924808211277794, 4.401385098288736389005869059414, 4.81391826904590205433891949191, 6.83627299767424762434004334103, 7.198091122477037674034007597481, 8.500563837153231671012271635689, 8.94276919870676128380362227109, 9.961518417239224837478840089749, 10.6076160897532778855483325892, 11.511066887756831719903685932358, 11.8889738347483923844544106396, 12.89056669318867564771363792329, 14.339506192394108709780202385226, 14.92057036403021333534323379641, 16.006995810381657985639473838535, 16.54213692470065105305775221478, 17.17676970887067444725659240656, 17.848447318224984343647927473173, 19.11834934446059226336284837574, 19.73622221970667487313870692144, 20.42196312463596094715782490663, 20.82934828471935521707934275932

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