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Properties

Label 1-967-967.441-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.0834 + 0.996i$
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  + (−0.791 + 0.610i)2-s + (0.494 + 0.869i)3-s + (0.254 − 0.967i)4-s + (−0.687 − 0.726i)5-s + (−0.922 − 0.386i)6-s + (0.483 + 0.875i)7-s + (0.389 + 0.921i)8-s + (−0.511 + 0.859i)9-s + (0.987 + 0.155i)10-s + (0.833 − 0.552i)11-s + (0.966 − 0.257i)12-s + (0.0617 − 0.998i)13-s + (−0.917 − 0.398i)14-s + (0.291 − 0.956i)15-s + (−0.870 − 0.491i)16-s + (−0.932 + 0.362i)17-s + ⋯
L(s)  = 1  + (−0.791 + 0.610i)2-s + (0.494 + 0.869i)3-s + (0.254 − 0.967i)4-s + (−0.687 − 0.726i)5-s + (−0.922 − 0.386i)6-s + (0.483 + 0.875i)7-s + (0.389 + 0.921i)8-s + (−0.511 + 0.859i)9-s + (0.987 + 0.155i)10-s + (0.833 − 0.552i)11-s + (0.966 − 0.257i)12-s + (0.0617 − 0.998i)13-s + (−0.917 − 0.398i)14-s + (0.291 − 0.956i)15-s + (−0.870 − 0.491i)16-s + (−0.932 + 0.362i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0834 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0834 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.0834 + 0.996i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.0834 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7218666116 + 0.7848465641i\)
\(L(\frac12)\) \(\approx\) \(0.7218666116 + 0.7848465641i\)
\(L(1)\) \(\approx\) \(0.7403856836 + 0.3958545782i\)
\(L(1)\) \(\approx\) \(0.7403856836 + 0.3958545782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.791 + 0.610i)T \)
3 \( 1 + (0.494 + 0.869i)T \)
5 \( 1 + (-0.687 - 0.726i)T \)
7 \( 1 + (0.483 + 0.875i)T \)
11 \( 1 + (0.833 - 0.552i)T \)
13 \( 1 + (0.0617 - 0.998i)T \)
17 \( 1 + (-0.932 + 0.362i)T \)
19 \( 1 + (0.228 - 0.973i)T \)
23 \( 1 + (-0.107 + 0.994i)T \)
29 \( 1 + (0.951 - 0.307i)T \)
31 \( 1 + (-0.197 + 0.980i)T \)
37 \( 1 + (-0.986 + 0.161i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (0.975 + 0.219i)T \)
47 \( 1 + (0.867 + 0.497i)T \)
53 \( 1 + (0.983 + 0.181i)T \)
59 \( 1 + (-0.395 - 0.918i)T \)
61 \( 1 + (0.0227 - 0.999i)T \)
67 \( 1 + (-0.750 + 0.660i)T \)
71 \( 1 + (0.924 - 0.380i)T \)
73 \( 1 + (0.983 - 0.181i)T \)
79 \( 1 + (0.471 - 0.881i)T \)
83 \( 1 + (0.998 - 0.0520i)T \)
89 \( 1 + (0.947 + 0.319i)T \)
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

−21.18863898123189085146385669164, −20.427762628253031492096943859349, −19.83254985878335349739080213214, −19.25290257705006280492727329183, −18.47623092202300706294048615014, −17.89052697332717311197065986419, −17.07994041813687660901574090112, −16.25483850643915413755866224501, −15.083820373131231579288871656403, −14.18539055135948091653088652224, −13.69414229911598397203482208054, −12.3525625944485042589635254783, −11.937698190555417403093882606606, −11.102950967643650836136862580448, −10.33842072041169705645093631148, −9.22591228635628801335569741430, −8.500715242246061674956819680134, −7.578562386812500367537912950521, −7.04014369997188050459373504972, −6.45494221131176267051881278255, −4.19017453427716270487896553137, −3.88648987713847313715280131030, −2.56426629232200281790359837262, −1.82863886835150150315385033638, −0.714479860455717605565065783569, 1.00251884556287367170639709466, 2.31298512064925327034761957225, 3.48961518066058756801854424131, 4.692142567429590575735621821533, 5.2666492120559872688770141951, 6.24839668798607446594687197748, 7.610043717077688443855969198676, 8.287888863647109109983793528233, 8.98738990073267617527959901633, 9.30131864767858761049337881446, 10.65898628738800404613778513091, 11.23301551352579097851331050591, 12.11876978165246493668843937160, 13.44757878036506119446251013198, 14.330241793721665647171429492286, 15.253214955745253377647812688, 15.64630067403914740244593856307, 16.15281005276163386256784747710, 17.357157100056621355527714049586, 17.662212426137725979232474742568, 19.02987490486768922824119812560, 19.6885207858997833756516523087, 20.04883921333950524811454754155, 21.06858648309633330186932502889, 21.86907357369962585650945604307

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