Properties

Label 1-967-967.30-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.948 + 0.317i$
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  + (0.938 − 0.344i)2-s + (0.592 + 0.805i)3-s + (0.763 − 0.646i)4-s + (−0.668 − 0.744i)5-s + (0.833 + 0.552i)6-s + (−0.945 + 0.325i)7-s + (0.494 − 0.869i)8-s + (−0.297 + 0.954i)9-s + (−0.883 − 0.468i)10-s + (−0.864 − 0.502i)11-s + (0.972 + 0.232i)12-s + (−0.864 + 0.502i)13-s + (−0.775 + 0.631i)14-s + (0.203 − 0.979i)15-s + (0.165 − 0.986i)16-s + (−0.999 − 0.0195i)17-s + ⋯
L(s)  = 1  + (0.938 − 0.344i)2-s + (0.592 + 0.805i)3-s + (0.763 − 0.646i)4-s + (−0.668 − 0.744i)5-s + (0.833 + 0.552i)6-s + (−0.945 + 0.325i)7-s + (0.494 − 0.869i)8-s + (−0.297 + 0.954i)9-s + (−0.883 − 0.468i)10-s + (−0.864 − 0.502i)11-s + (0.972 + 0.232i)12-s + (−0.864 + 0.502i)13-s + (−0.775 + 0.631i)14-s + (0.203 − 0.979i)15-s + (0.165 − 0.986i)16-s + (−0.999 − 0.0195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.948 + 0.317i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.948 + 0.317i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03045503110 + 0.1868157884i\)
\(L(\frac12)\) \(\approx\) \(0.03045503110 + 0.1868157884i\)
\(L(1)\) \(\approx\) \(1.194627041 - 0.03621454435i\)
\(L(1)\) \(\approx\) \(1.194627041 - 0.03621454435i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.938 - 0.344i)T \)
3 \( 1 + (0.592 + 0.805i)T \)
5 \( 1 + (-0.668 - 0.744i)T \)
7 \( 1 + (-0.945 + 0.325i)T \)
11 \( 1 + (-0.864 - 0.502i)T \)
13 \( 1 + (-0.864 + 0.502i)T \)
17 \( 1 + (-0.999 - 0.0195i)T \)
19 \( 1 + (-0.260 + 0.965i)T \)
23 \( 1 + (-0.967 + 0.250i)T \)
29 \( 1 + (0.560 - 0.828i)T \)
31 \( 1 + (-0.957 - 0.288i)T \)
37 \( 1 + (-0.145 + 0.989i)T \)
41 \( 1 + (-0.990 - 0.136i)T \)
43 \( 1 + (0.909 + 0.416i)T \)
47 \( 1 + (0.892 - 0.451i)T \)
53 \( 1 + (0.460 + 0.887i)T \)
59 \( 1 + (-0.407 - 0.913i)T \)
61 \( 1 + (-0.990 - 0.136i)T \)
67 \( 1 + (-0.957 + 0.288i)T \)
71 \( 1 + (0.938 + 0.344i)T \)
73 \( 1 + (0.460 - 0.887i)T \)
79 \( 1 + (-0.0292 + 0.999i)T \)
83 \( 1 + (0.353 - 0.935i)T \)
89 \( 1 + (-0.957 + 0.288i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−21.66497500475531262162443976714, −20.32769606537409630693719946003, −19.88548614660117203332134889851, −19.38631319259783335377239443329, −18.16148238843051454260741151346, −17.62724399648337263636568960713, −16.375626997848534430877640119027, −15.46796705754880582563149461991, −15.13209944314182355079270014087, −14.13499742341185721108846715259, −13.48698143035634725565219981955, −12.55256552827345435234208368413, −12.32602822564579117768012629753, −11.03918675623337087821052205211, −10.29847867980190324505150804650, −8.95750133624908197100210530402, −7.90926328453884654639184629988, −7.134175174206352750636340981787, −6.87652477529830727903411094367, −5.81014807620029397136802938540, −4.5419225662580162642995342127, −3.61552295442723936387034279999, −2.74038631789953134037087633149, −2.25545531569780713943215040043, −0.04635679892608956613848513365, 1.98795480730249657879418342740, 2.82533764812791941052319135786, 3.76382504012765339645112353161, 4.39138263007985837778991698427, 5.270890834690286807759196781719, 6.117898193535381762963582876860, 7.40184845344416547408017584019, 8.300732107948059577565971730571, 9.303274603029407837941870914727, 10.01921990583619142628538403522, 10.86568820467366239200793096075, 11.870726836369044740387797647936, 12.52878118953152704145759483727, 13.41647365263552693697425350745, 13.99606777222732031532209837731, 15.25463518291467545988332912506, 15.49129226817654165004759361788, 16.32581619879972923422321836180, 16.82316504492067088163855322380, 18.68143526871423095776398647835, 19.29054927705093429862652107152, 19.98520979474607472742957076317, 20.50572730388701824575864488502, 21.41516024589739766297100307992, 21.96440195446590527474845403448

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