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Properties

Label 1-1007-1007.453-r0-0-0
Degree $1$
Conductor $1007$
Sign $0.568 + 0.822i$
Analytic cond. $4.67648$
Root an. cond. $4.67648$
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  + (0.783 + 0.621i)2-s + (0.897 − 0.440i)3-s + (0.226 + 0.974i)4-s + (0.673 + 0.739i)5-s + (0.977 + 0.213i)6-s + (−0.200 − 0.979i)7-s + (−0.428 + 0.903i)8-s + (0.611 − 0.791i)9-s + (0.0670 + 0.997i)10-s + (0.987 − 0.160i)11-s + (0.632 + 0.774i)12-s + (−0.545 + 0.837i)13-s + (0.452 − 0.891i)14-s + (0.930 + 0.367i)15-s + (−0.897 + 0.440i)16-s + (0.909 + 0.416i)17-s + ⋯
L(s)  = 1  + (0.783 + 0.621i)2-s + (0.897 − 0.440i)3-s + (0.226 + 0.974i)4-s + (0.673 + 0.739i)5-s + (0.977 + 0.213i)6-s + (−0.200 − 0.979i)7-s + (−0.428 + 0.903i)8-s + (0.611 − 0.791i)9-s + (0.0670 + 0.997i)10-s + (0.987 − 0.160i)11-s + (0.632 + 0.774i)12-s + (−0.545 + 0.837i)13-s + (0.452 − 0.891i)14-s + (0.930 + 0.367i)15-s + (−0.897 + 0.440i)16-s + (0.909 + 0.416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1007\)    =    \(19 \cdot 53\)
Sign: $0.568 + 0.822i$
Analytic conductor: \(4.67648\)
Root analytic conductor: \(4.67648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1007} (453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1007,\ (0:\ ),\ 0.568 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.326928155 + 1.743844236i\)
\(L(\frac12)\) \(\approx\) \(3.326928155 + 1.743844236i\)
\(L(1)\) \(\approx\) \(2.260434557 + 0.7839577289i\)
\(L(1)\) \(\approx\) \(2.260434557 + 0.7839577289i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (0.783 + 0.621i)T \)
3 \( 1 + (0.897 - 0.440i)T \)
5 \( 1 + (0.673 + 0.739i)T \)
7 \( 1 + (-0.200 - 0.979i)T \)
11 \( 1 + (0.987 - 0.160i)T \)
13 \( 1 + (-0.545 + 0.837i)T \)
17 \( 1 + (0.909 + 0.416i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.982 + 0.186i)T \)
31 \( 1 + (-0.987 - 0.160i)T \)
37 \( 1 + (-0.970 - 0.239i)T \)
41 \( 1 + (-0.329 - 0.944i)T \)
43 \( 1 + (0.994 + 0.107i)T \)
47 \( 1 + (-0.303 - 0.952i)T \)
59 \( 1 + (0.611 + 0.791i)T \)
61 \( 1 + (0.815 + 0.579i)T \)
67 \( 1 + (0.956 + 0.291i)T \)
71 \( 1 + (-0.994 - 0.107i)T \)
73 \( 1 + (0.0938 + 0.995i)T \)
79 \( 1 + (0.999 + 0.0268i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.964 - 0.265i)T \)
97 \( 1 + (-0.303 + 0.952i)T \)
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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

−21.457325832397842805356370694967, −20.74256224636576920363127739542, −20.243589151672651394605963590980, −19.3563606073502580701112968900, −18.8180112773021279984321479928, −17.68071084861509854241756460954, −16.58718542668874462216995739316, −15.786384053605800182760176265388, −14.86389550250897514587301610658, −14.499253530711581325082540396701, −13.52894479211726524401266322320, −12.75606367206700356337183880324, −12.27056390491162225779927245654, −11.18802007143564082286138592232, −10.0610862867138297341994757638, −9.40318136993052175130953539385, −9.034234831527928222422747547244, −7.784392687719028480470675832433, −6.53281601613117991052750117764, −5.3354547610019123531753145511, −5.09449018080973227325104522155, −3.81206060081763084193499366587, −3.022167224512388389431078483519, −2.14449908771886623366207899010, −1.27721282120242529785762612031, 1.50305760348730109955338349195, 2.457939920212247423661966053936, 3.60419763502491547744030794820, 3.90191803948637337096597924160, 5.34557773975518909198755821804, 6.43185632587129559108362222996, 7.07760177238446252452500416830, 7.444600772218683035318542232650, 8.76453464428446415069675023146, 9.45277871640875007868451263986, 10.50802306179784524126537928935, 11.57617609660269255869566789590, 12.55249336515563173746471947831, 13.31452971822976084027524378872, 14.01267790499948402369840558667, 14.55503947289354155413645008803, 14.91956078503657626994110592411, 16.31101771447701554622476369821, 17.00696744753279165205636440192, 17.60893841568099771572259441138, 18.80693178424428880880941357503, 19.34991092482803720977560645820, 20.40569154094601382077594863309, 21.03659049423693438848927832403, 21.82740344764018185103435968295

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