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Properties

Label 1-1009-1009.137-r0-0-0
Degree $1$
Conductor $1009$
Sign $-0.594 + 0.804i$
Analytic cond. $4.68577$
Root an. cond. $4.68577$
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  + (0.603 + 0.797i)2-s + (−0.680 + 0.733i)3-s + (−0.270 + 0.962i)4-s + (0.811 + 0.583i)5-s + (−0.995 − 0.0995i)6-s + (−0.456 − 0.889i)7-s + (−0.930 + 0.365i)8-s + (−0.0747 − 0.997i)9-s + (0.0249 + 0.999i)10-s + (0.986 − 0.161i)11-s + (−0.521 − 0.853i)12-s + (0.258 − 0.965i)13-s + (0.433 − 0.900i)14-s + (−0.980 + 0.198i)15-s + (−0.853 − 0.521i)16-s + (0.234 + 0.972i)17-s + ⋯
L(s)  = 1  + (0.603 + 0.797i)2-s + (−0.680 + 0.733i)3-s + (−0.270 + 0.962i)4-s + (0.811 + 0.583i)5-s + (−0.995 − 0.0995i)6-s + (−0.456 − 0.889i)7-s + (−0.930 + 0.365i)8-s + (−0.0747 − 0.997i)9-s + (0.0249 + 0.999i)10-s + (0.986 − 0.161i)11-s + (−0.521 − 0.853i)12-s + (0.258 − 0.965i)13-s + (0.433 − 0.900i)14-s + (−0.980 + 0.198i)15-s + (−0.853 − 0.521i)16-s + (0.234 + 0.972i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1009\)
Sign: $-0.594 + 0.804i$
Analytic conductor: \(4.68577\)
Root analytic conductor: \(4.68577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1009} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1009,\ (0:\ ),\ -0.594 + 0.804i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8361336040 + 1.658060273i\)
\(L(\frac12)\) \(\approx\) \(0.8361336040 + 1.658060273i\)
\(L(1)\) \(\approx\) \(0.9944684687 + 0.8849076254i\)
\(L(1)\) \(\approx\) \(0.9944684687 + 0.8849076254i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1009 \( 1 \)
good2 \( 1 + (0.603 + 0.797i)T \)
3 \( 1 + (-0.680 + 0.733i)T \)
5 \( 1 + (0.811 + 0.583i)T \)
7 \( 1 + (-0.456 - 0.889i)T \)
11 \( 1 + (0.986 - 0.161i)T \)
13 \( 1 + (0.258 - 0.965i)T \)
17 \( 1 + (0.234 + 0.972i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (0.804 + 0.593i)T \)
29 \( 1 + (-0.0498 - 0.998i)T \)
31 \( 1 + (-0.990 + 0.136i)T \)
37 \( 1 + (0.642 + 0.766i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.258 + 0.965i)T \)
47 \( 1 + (0.884 + 0.467i)T \)
53 \( 1 + (0.376 + 0.926i)T \)
59 \( 1 + (-0.330 + 0.943i)T \)
61 \( 1 + (0.0373 - 0.999i)T \)
67 \( 1 + (0.456 - 0.889i)T \)
71 \( 1 + (-0.603 - 0.797i)T \)
73 \( 1 + (0.982 - 0.185i)T \)
79 \( 1 + (0.833 + 0.552i)T \)
83 \( 1 + (0.689 + 0.724i)T \)
89 \( 1 + (-0.633 - 0.773i)T \)
97 \( 1 + (0.859 - 0.510i)T \)
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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

−21.61340597396899832934560895619, −20.57558302939284008904349054857, −19.91512242861558612003675834365, −18.82186602379509415223814992008, −18.51683259552825368944947131581, −17.673574359330373485582014165949, −16.58974585376398385950145128432, −16.09203810651828484175764486349, −14.65716694978653742878810482007, −13.98214099307615366542773354205, −13.26426232760959238640327067763, −12.45796868446383095122799051184, −11.96201907362900781409334900327, −11.30647134015155152422344038177, −10.18841221464234307583985587264, −9.20232921000955315348354920819, −8.887421625993688018479749972626, −7.0317427890370640394344482038, −6.418196675982761776455901104996, −5.43211464527446489070260299818, −5.09537471103368104728506232363, −3.76970190431164336683317791199, −2.49211644201434670427587571214, −1.76705808180215413307178169400, −0.86737737065182902087331477576, 1.12458836792026233036836021474, 3.12825448504352791466169337825, 3.56434558982970308249076321341, 4.59210943374152147678009409117, 5.621905422121055114494018931341, 6.18070789981537487160081053881, 6.875788123461111538731968647466, 7.83589789737106968967156486027, 9.20072101206396064150873047170, 9.76147709036068294748842997647, 10.73970188515494670375650805353, 11.444731373230523716121458287081, 12.55989908887755944386853446747, 13.37217234165020803432522762195, 14.05151683449083656254108396764, 14.98910142700138108070511595348, 15.43699567664499295055471285080, 16.68247738719468720571361773531, 16.9238442380630996407146008877, 17.64280762827973193996998256254, 18.385011996413869208936335897642, 19.778527069852595395514006327825, 20.62745115177423174414897483936, 21.49288743519630725560286071619, 22.08584277123890760098553480444

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