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Properties

Label 1-1007-1007.329-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: $\chi_{1007} (329, \cdot )$
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

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

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