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Properties

Label 1-1012-1012.267-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.728 + 0.685i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (0.466 + 0.884i)3-s + (0.564 + 0.825i)5-s + (0.0855 + 0.996i)7-s + (−0.564 + 0.825i)9-s + (0.516 + 0.856i)13-s + (−0.466 + 0.884i)15-s + (0.736 − 0.676i)17-s + (0.198 + 0.980i)19-s + (−0.841 + 0.540i)21-s + (−0.362 + 0.931i)25-s + (−0.993 − 0.113i)27-s + (0.198 − 0.980i)29-s + (0.985 + 0.170i)31-s + (−0.774 + 0.633i)35-s + (−0.610 − 0.791i)37-s + ⋯
L(s)  = 1  + (0.466 + 0.884i)3-s + (0.564 + 0.825i)5-s + (0.0855 + 0.996i)7-s + (−0.564 + 0.825i)9-s + (0.516 + 0.856i)13-s + (−0.466 + 0.884i)15-s + (0.736 − 0.676i)17-s + (0.198 + 0.980i)19-s + (−0.841 + 0.540i)21-s + (−0.362 + 0.931i)25-s + (−0.993 − 0.113i)27-s + (0.198 − 0.980i)29-s + (0.985 + 0.170i)31-s + (−0.774 + 0.633i)35-s + (−0.610 − 0.791i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.728 + 0.685i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.728 + 0.685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7445009042 + 1.877093800i\)
\(L(\frac12)\) \(\approx\) \(0.7445009042 + 1.877093800i\)
\(L(1)\) \(\approx\) \(1.118976389 + 0.8528268471i\)
\(L(1)\) \(\approx\) \(1.118976389 + 0.8528268471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.466 + 0.884i)T \)
5 \( 1 + (0.564 + 0.825i)T \)
7 \( 1 + (0.0855 + 0.996i)T \)
13 \( 1 + (0.516 + 0.856i)T \)
17 \( 1 + (0.736 - 0.676i)T \)
19 \( 1 + (0.198 + 0.980i)T \)
29 \( 1 + (0.198 - 0.980i)T \)
31 \( 1 + (0.985 + 0.170i)T \)
37 \( 1 + (-0.610 - 0.791i)T \)
41 \( 1 + (0.610 - 0.791i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.921 - 0.389i)T \)
59 \( 1 + (-0.974 + 0.226i)T \)
61 \( 1 + (-0.897 + 0.441i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.774 - 0.633i)T \)
73 \( 1 + (-0.870 + 0.491i)T \)
79 \( 1 + (0.516 + 0.856i)T \)
83 \( 1 + (-0.0285 - 0.999i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.0285 - 0.999i)T \)
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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

−21.05299973315217511757166557815, −20.39294250539355205738962661961, −19.93840447686334256963462280639, −19.085690510101927771305574498808, −18.08414399321828484943983209629, −17.42104475270340188820572536305, −16.92040045215363791143741052828, −15.85152233702531219668405088953, −14.872033179599743774880294868966, −13.85867841348166528481947969638, −13.48723272611556772424129256331, −12.73419399243225861394401348472, −12.03792318935186931262800637826, −10.835840235410051292104762425005, −10.03552930043826403754720493782, −9.02901333106813619006400992929, −8.2813593468763943934201349653, −7.583095481935213346791556661864, −6.61366544697411352910396617383, −5.779397111938388429644778239562, −4.77488334247497984840814847099, −3.63938416268054339836646489664, −2.693135054062989532803253834106, −1.38735243551203797938142793203, −0.87632377007382117760586009528, 1.77419730919020989589533284504, 2.622810816897261240944170775117, 3.39997727614453829617289664831, 4.42239843362633225554268747022, 5.56536632324841626246969925314, 6.07848973122107858120644761777, 7.321366211098194686499290614216, 8.313405505536100125157800259093, 9.17372494892038342246333242750, 9.79271332557612425223912225690, 10.55385334136736813920680094392, 11.48265751703574915982714313806, 12.17323810457900020255728442346, 13.60183559595499702910523919420, 14.118679574546679788987413654690, 14.78582052146215589409553003253, 15.592363362633431798355979425013, 16.25026564901951520409440752640, 17.160776148760716374995604643275, 18.18872594886363230156635166835, 18.82742151312937920391912871975, 19.42586673314068081316835095402, 20.7353073232149323142916395333, 21.23400404630537640215809105441, 21.647162668099358610442196724864

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