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Properties

Label 1-967-967.84-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.512 + 0.858i$
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  + (−0.285 + 0.958i)2-s + (0.811 + 0.584i)3-s + (−0.837 − 0.547i)4-s + (0.847 + 0.530i)5-s + (−0.791 + 0.610i)6-s + (0.975 − 0.219i)7-s + (0.763 − 0.646i)8-s + (0.316 + 0.948i)9-s + (−0.750 + 0.660i)10-s + (0.938 + 0.344i)11-s + (−0.359 − 0.933i)12-s + (−0.767 − 0.641i)13-s + (−0.0682 + 0.997i)14-s + (0.377 + 0.926i)15-s + (0.401 + 0.915i)16-s + (−0.511 + 0.859i)17-s + ⋯
L(s)  = 1  + (−0.285 + 0.958i)2-s + (0.811 + 0.584i)3-s + (−0.837 − 0.547i)4-s + (0.847 + 0.530i)5-s + (−0.791 + 0.610i)6-s + (0.975 − 0.219i)7-s + (0.763 − 0.646i)8-s + (0.316 + 0.948i)9-s + (−0.750 + 0.660i)10-s + (0.938 + 0.344i)11-s + (−0.359 − 0.933i)12-s + (−0.767 − 0.641i)13-s + (−0.0682 + 0.997i)14-s + (0.377 + 0.926i)15-s + (0.401 + 0.915i)16-s + (−0.511 + 0.859i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.512 + 0.858i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.512 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063065632 + 1.873179589i\)
\(L(\frac12)\) \(\approx\) \(1.063065632 + 1.873179589i\)
\(L(1)\) \(\approx\) \(1.106977094 + 0.9513917611i\)
\(L(1)\) \(\approx\) \(1.106977094 + 0.9513917611i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.285 + 0.958i)T \)
3 \( 1 + (0.811 + 0.584i)T \)
5 \( 1 + (0.847 + 0.530i)T \)
7 \( 1 + (0.975 - 0.219i)T \)
11 \( 1 + (0.938 + 0.344i)T \)
13 \( 1 + (-0.767 - 0.641i)T \)
17 \( 1 + (-0.511 + 0.859i)T \)
19 \( 1 + (0.643 - 0.765i)T \)
23 \( 1 + (-0.638 - 0.769i)T \)
29 \( 1 + (0.126 + 0.991i)T \)
31 \( 1 + (-0.322 - 0.946i)T \)
37 \( 1 + (0.413 + 0.910i)T \)
41 \( 1 + (-0.576 + 0.816i)T \)
43 \( 1 + (-0.235 - 0.971i)T \)
47 \( 1 + (-0.741 - 0.670i)T \)
53 \( 1 + (0.746 + 0.665i)T \)
59 \( 1 + (-0.961 + 0.276i)T \)
61 \( 1 + (0.995 + 0.0909i)T \)
67 \( 1 + (0.981 - 0.193i)T \)
71 \( 1 + (0.972 + 0.232i)T \)
73 \( 1 + (0.746 - 0.665i)T \)
79 \( 1 + (0.483 + 0.875i)T \)
83 \( 1 + (-0.971 - 0.238i)T \)
89 \( 1 + (-0.658 - 0.752i)T \)
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

−21.38288873971351789999577522514, −20.5954765374787965905466923012, −19.96712537473013805198189776673, −19.293529431797263814784312732235, −18.337281077920684370663250834027, −17.79231207314148445002794082597, −17.16558941106420173292954995400, −16.14158383219970650202107526151, −14.62041151474683342629419119490, −14.02971694043466412297916214036, −13.67807415829918838425178001952, −12.52119971771761066678525673421, −11.936104151810488377963627227324, −11.24022204388695775011550130822, −9.73317575912918918067780560401, −9.46702330119676357613383714566, −8.599348443400707302452123585911, −7.90342619220094214908516621747, −6.870127823448833844779562117921, −5.55672230325220332592751047338, −4.581106489475707481216093272010, −3.64886193867968278180201382232, −2.405448812410567914314796444421, −1.82252864517870666170036583329, −1.04998647190110653967603586157, 1.43698168157487207926498067948, 2.3701587315041425459310760743, 3.72772626427879137669337178453, 4.68295131398428079792912255550, 5.345024142671828894157593649137, 6.54285708089864471652735318005, 7.30090485916664636706966674256, 8.20239378222283862517437001895, 8.92093034440459319297696396156, 9.80575685296250415791477815707, 10.31531392394970108760448610687, 11.25547363489372805489560197927, 12.7983036734764308661550720779, 13.7365336848471622384923742454, 14.25770251261180403492668677358, 15.02871841765795553256896385816, 15.23751586249259396773397936873, 16.72499305791917044121297739251, 17.12152854816674035355922372606, 18.00624700427203634556752565460, 18.59279097207663567910734007776, 19.963219751444453803072028414157, 20.06643727285450516589357442221, 21.487195610254036948464270491055, 22.08033297326865338425104219137

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