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Properties

Label 1-967-967.742-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.345 + 0.938i$
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  + (−0.247 + 0.968i)2-s + (−0.203 + 0.979i)3-s + (−0.877 − 0.480i)4-s + (0.998 − 0.0455i)5-s + (−0.898 − 0.439i)6-s + (−0.995 − 0.0909i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (−0.203 + 0.979i)10-s + (−0.917 − 0.398i)11-s + (0.648 − 0.761i)12-s + (−0.803 − 0.595i)13-s + (0.334 − 0.942i)14-s + (−0.158 + 0.987i)15-s + (0.538 + 0.842i)16-s + (−0.775 + 0.631i)17-s + ⋯
L(s)  = 1  + (−0.247 + 0.968i)2-s + (−0.203 + 0.979i)3-s + (−0.877 − 0.480i)4-s + (0.998 − 0.0455i)5-s + (−0.898 − 0.439i)6-s + (−0.995 − 0.0909i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (−0.203 + 0.979i)10-s + (−0.917 − 0.398i)11-s + (0.648 − 0.761i)12-s + (−0.803 − 0.595i)13-s + (0.334 − 0.942i)14-s + (−0.158 + 0.987i)15-s + (0.538 + 0.842i)16-s + (−0.775 + 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.345 + 0.938i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (742, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.345 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3344644561 + 0.4797473099i\)
\(L(\frac12)\) \(\approx\) \(0.3344644561 + 0.4797473099i\)
\(L(1)\) \(\approx\) \(0.4979183097 + 0.3596930944i\)
\(L(1)\) \(\approx\) \(0.4979183097 + 0.3596930944i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.247 + 0.968i)T \)
3 \( 1 + (-0.203 + 0.979i)T \)
5 \( 1 + (0.998 - 0.0455i)T \)
7 \( 1 + (-0.995 - 0.0909i)T \)
11 \( 1 + (-0.917 - 0.398i)T \)
13 \( 1 + (-0.803 - 0.595i)T \)
17 \( 1 + (-0.775 + 0.631i)T \)
19 \( 1 + (-0.746 - 0.665i)T \)
23 \( 1 + (-0.854 + 0.519i)T \)
29 \( 1 + (0.917 - 0.398i)T \)
31 \( 1 + (-0.974 - 0.225i)T \)
37 \( 1 + (-0.803 - 0.595i)T \)
41 \( 1 + (0.0682 + 0.997i)T \)
43 \( 1 + (-0.934 + 0.356i)T \)
47 \( 1 + (0.158 - 0.987i)T \)
53 \( 1 + (0.0227 + 0.999i)T \)
59 \( 1 + (0.0227 - 0.999i)T \)
61 \( 1 + (-0.829 + 0.557i)T \)
67 \( 1 + (-0.682 - 0.730i)T \)
71 \( 1 + (0.962 + 0.269i)T \)
73 \( 1 + (0.0227 - 0.999i)T \)
79 \( 1 + (-0.0227 + 0.999i)T \)
83 \( 1 + (0.898 + 0.439i)T \)
89 \( 1 + (-0.291 + 0.956i)T \)
97 \( 1 + T \)
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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

−21.289709592239063961082337766771, −20.36724939979271120547852098380, −19.71744311246606161023883762815, −18.82908293710671384500130166721, −18.36111436286542556448837553815, −17.625643622783130518785897423671, −16.92571808832282231014744050212, −16.0958107960608252602654297686, −14.52180494315808824488710836717, −13.80137194734451049612202044835, −13.122213062290090003848606910474, −12.5057522422021644411704303544, −11.940235739887053988771568370702, −10.639848429698179383434112580955, −10.16036155085524485980430418552, −9.219270080104231019061574218154, −8.47346594907026935238500232147, −7.2842988520254515251401023085, −6.54941248283404536854538105366, −5.51637453948976944945859877186, −4.6222218123078978752330563418, −3.12069805319294353882723393153, −2.29548469473316346205947655557, −1.81883462730817883952043516751, −0.31946814617145970789085141895, 0.394332119022854564530300344635, 2.288447562477702205401527003433, 3.39992694827844055151247970039, 4.55670251789481445967737631758, 5.352496270678419939052919518164, 6.04980867237039157165727636433, 6.72185425089434170961675975163, 8.01508191726271927517492500272, 8.9237956755887251934244953094, 9.60539072335516838070965762842, 10.296486631123441125124167874693, 10.7762632428658958009201170810, 12.47920182841547836935296634514, 13.27786062059161436520763346860, 13.89397800000996607003909441706, 15.0150313611974004853476385280, 15.47541489953947863025335424781, 16.318556240838975936155866110690, 16.95009135488346274157932929396, 17.6278100792483643250807453568, 18.289651524482729344805430277230, 19.53525178575087964148700255723, 20.05440725931280239835536304574, 21.50486695665535746468137023948, 21.75445333250888601827728801937

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