Properties

Label 1-967-967.640-r1-0-0
Degree $1$
Conductor $967$
Sign $0.890 + 0.454i$
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  + (0.993 − 0.116i)2-s + (−0.951 − 0.307i)3-s + (0.972 − 0.232i)4-s + (−0.241 − 0.970i)5-s + (−0.981 − 0.193i)6-s + (−0.592 − 0.805i)7-s + (0.938 − 0.344i)8-s + (0.811 + 0.584i)9-s + (−0.353 − 0.935i)10-s + (−0.984 − 0.174i)11-s + (−0.996 − 0.0779i)12-s + (0.984 − 0.174i)13-s + (−0.682 − 0.730i)14-s + (−0.0682 + 0.997i)15-s + (0.892 − 0.451i)16-s + (0.494 + 0.869i)17-s + ⋯
L(s)  = 1  + (0.993 − 0.116i)2-s + (−0.951 − 0.307i)3-s + (0.972 − 0.232i)4-s + (−0.241 − 0.970i)5-s + (−0.981 − 0.193i)6-s + (−0.592 − 0.805i)7-s + (0.938 − 0.344i)8-s + (0.811 + 0.584i)9-s + (−0.353 − 0.935i)10-s + (−0.984 − 0.174i)11-s + (−0.996 − 0.0779i)12-s + (0.984 − 0.174i)13-s + (−0.682 − 0.730i)14-s + (−0.0682 + 0.997i)15-s + (0.892 − 0.451i)16-s + (0.494 + 0.869i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.890 + 0.454i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (640, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.890 + 0.454i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.709198655 + 0.4108113193i\)
\(L(\frac12)\) \(\approx\) \(1.709198655 + 0.4108113193i\)
\(L(1)\) \(\approx\) \(1.214909493 - 0.3381974413i\)
\(L(1)\) \(\approx\) \(1.214909493 - 0.3381974413i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.993 + 0.116i)T \)
3 \( 1 + (0.951 + 0.307i)T \)
5 \( 1 + (0.241 + 0.970i)T \)
7 \( 1 + (0.592 + 0.805i)T \)
11 \( 1 + (0.984 + 0.174i)T \)
13 \( 1 + (-0.984 + 0.174i)T \)
17 \( 1 + (-0.494 - 0.869i)T \)
19 \( 1 + (0.0876 - 0.996i)T \)
23 \( 1 + (0.425 - 0.905i)T \)
29 \( 1 + (-0.750 - 0.660i)T \)
31 \( 1 + (0.995 + 0.0974i)T \)
37 \( 1 + (0.0487 - 0.998i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (-0.371 - 0.928i)T \)
47 \( 1 + (0.987 - 0.155i)T \)
53 \( 1 + (0.775 - 0.631i)T \)
59 \( 1 + (-0.787 + 0.615i)T \)
61 \( 1 + (-0.460 - 0.887i)T \)
67 \( 1 + (-0.995 + 0.0974i)T \)
71 \( 1 + (-0.993 - 0.116i)T \)
73 \( 1 + (0.775 + 0.631i)T \)
79 \( 1 + (0.00975 - 0.999i)T \)
83 \( 1 + (0.799 + 0.600i)T \)
89 \( 1 + (-0.995 + 0.0974i)T \)
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

−21.67714524659726211831617356304, −21.12677310978420703520604231112, −20.16241951462957023967851451114, −18.95512570654039397714624411107, −18.369880570325281348462127259683, −17.615376695889546609687864118973, −16.13015557044540341748595436310, −16.04431902974534241846646392003, −15.28790041018898033930934128169, −14.462047746492361462797716142327, −13.39672466916885082709332718379, −12.67777837166298637805425045393, −11.83036975085055140867892189303, −11.19343762757326521963797538670, −10.51420550584873471345871059164, −9.64441325907008808017798289627, −8.2041558775198435490207572621, −7.04841802548203287424308286066, −6.512227798661860535271268557, −5.70213605460914911014347663849, −4.97638531166027069787876546958, −3.89021316779194542545903683099, −3.01817848980783177794542967221, −2.17464103602585264193471626607, −0.32390519725021893299829490205, 1.00120574424064547815905360942, 1.67768029712287372898904403341, 3.40047968101639165298476641420, 4.0403055941841953004140601512, 5.08228737359922525161657146515, 5.74316563027055462775801146994, 6.45340552998689361802206340068, 7.577980919944452507525295452477, 8.18541528457413520114254038713, 9.897186835833382669203025349851, 10.527310009939286900883889710665, 11.29542940774618231034414818308, 12.23014029868864408107123818152, 12.93176802561585774226175695142, 13.232453092651256763726946526280, 14.19371306583595597372844574126, 15.59747599622920711176762438830, 16.07592856392113124694466159007, 16.62307914517668637224859414359, 17.4221871892812591797860424668, 18.600050426262883772840094530150, 19.3900556845892048657573332719, 20.24389512126864055818421878601, 20.97148358225993487485799209159, 21.594314641422871758169558299970

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