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Properties

Label 1-967-967.149-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.380 + 0.924i$
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  + (0.779 − 0.626i)2-s + (−0.998 − 0.0585i)3-s + (0.216 − 0.976i)4-s + (−0.488 + 0.872i)5-s + (−0.815 + 0.579i)6-s + (0.341 + 0.940i)7-s + (−0.442 − 0.896i)8-s + (0.993 + 0.116i)9-s + (0.165 + 0.986i)10-s + (0.527 + 0.849i)11-s + (−0.272 + 0.962i)12-s + (−0.999 − 0.0325i)13-s + (0.854 + 0.519i)14-s + (0.538 − 0.842i)15-s + (−0.906 − 0.422i)16-s + (0.981 − 0.193i)17-s + ⋯
L(s)  = 1  + (0.779 − 0.626i)2-s + (−0.998 − 0.0585i)3-s + (0.216 − 0.976i)4-s + (−0.488 + 0.872i)5-s + (−0.815 + 0.579i)6-s + (0.341 + 0.940i)7-s + (−0.442 − 0.896i)8-s + (0.993 + 0.116i)9-s + (0.165 + 0.986i)10-s + (0.527 + 0.849i)11-s + (−0.272 + 0.962i)12-s + (−0.999 − 0.0325i)13-s + (0.854 + 0.519i)14-s + (0.538 − 0.842i)15-s + (−0.906 − 0.422i)16-s + (0.981 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.380 + 0.924i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.380 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3526154467 + 0.5266863246i\)
\(L(\frac12)\) \(\approx\) \(0.3526154467 + 0.5266863246i\)
\(L(1)\) \(\approx\) \(0.9076887889 - 0.03894919944i\)
\(L(1)\) \(\approx\) \(0.9076887889 - 0.03894919944i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.779 - 0.626i)T \)
3 \( 1 + (-0.998 - 0.0585i)T \)
5 \( 1 + (-0.488 + 0.872i)T \)
7 \( 1 + (0.341 + 0.940i)T \)
11 \( 1 + (0.527 + 0.849i)T \)
13 \( 1 + (-0.999 - 0.0325i)T \)
17 \( 1 + (0.981 - 0.193i)T \)
19 \( 1 + (-0.857 + 0.514i)T \)
23 \( 1 + (-0.822 + 0.568i)T \)
29 \( 1 + (-0.945 - 0.325i)T \)
31 \( 1 + (0.304 + 0.952i)T \)
37 \( 1 + (0.914 - 0.404i)T \)
41 \( 1 + (0.203 - 0.979i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (-0.851 + 0.525i)T \)
53 \( 1 + (-0.829 - 0.557i)T \)
59 \( 1 + (0.505 - 0.862i)T \)
61 \( 1 + (-0.949 + 0.313i)T \)
67 \( 1 + (-0.977 + 0.212i)T \)
71 \( 1 + (-0.932 + 0.362i)T \)
73 \( 1 + (-0.829 + 0.557i)T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (-0.0552 + 0.998i)T \)
89 \( 1 + (0.672 + 0.739i)T \)
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

−21.669711789111614360041175628896, −20.98225180226723159309168345141, −20.11780643943914410494581249436, −19.24967686018277181247342704426, −18.01439809236710194338223842589, −17.04411582780511273843982313624, −16.670588103794424026904124685068, −16.36309935004613844453254800041, −15.10705271829696413549364340636, −14.50330915597444609184349520110, −13.317913382644600240673451437733, −12.839678370083173583991451351104, −11.81478868114148518829180522800, −11.48466239007425187522812543404, −10.38279756670201440026470906615, −9.24826304091306985122126068448, −8.00578114289840209745104373000, −7.554615391700406877080603563, −6.44455397538902116808374491908, −5.75724349503384570209469715998, −4.58897050251245692466341030604, −4.433705841888760815334774267365, −3.32396044635036474488471631110, −1.56380730165557307183726290978, −0.23295407155860197258479145831, 1.62340221369353685368294296101, 2.3818423760164713323549197280, 3.62244074932152897750089423433, 4.48098337094154423453554028131, 5.362467147596528378607432753973, 6.11267724297567042322842685358, 6.971282945557492892628984316704, 7.81549163387832341978785505185, 9.505876015257120071898850356447, 10.10480453500604057552221332638, 10.95005656116992769108701035740, 11.888945612671171999304145160710, 12.06448521546012636568659674558, 12.830252185027451541492240475005, 14.24647164120915943963206679104, 14.800450316181377818205187723421, 15.40739734329963217225377696351, 16.314305403176667202877619476343, 17.49340847953386441310228849584, 18.17280538548800385594009731708, 19.01588248272879026459773492176, 19.43321592893435812566926223234, 20.643083443177228230274720320093, 21.519704488164451530164747474251, 22.10119843830521662047338438268

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