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Properties

Label 1-1007-1007.130-r0-0-0
Degree $1$
Conductor $1007$
Sign $0.975 + 0.221i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1007\)    =    \(19 \cdot 53\)
Sign: $0.975 + 0.221i$
Analytic conductor: \(4.67648\)
Root analytic conductor: \(4.67648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1007} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1007,\ (0:\ ),\ 0.975 + 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4664079170 + 0.05233868232i\)
\(L(\frac12)\) \(\approx\) \(0.4664079170 + 0.05233868232i\)
\(L(1)\) \(\approx\) \(0.4795855432 - 0.1044047424i\)
\(L(1)\) \(\approx\) \(0.4795855432 - 0.1044047424i\)

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

−22.08334604387444487901627673383, −20.666298924435785325045708245475, −19.50922183361271108376282072256, −19.08940681328565663306921967229, −18.447244913262228541838810134597, −17.70389970145094464659593047238, −17.07001563632810078015335498139, −16.07961978165071585287228123894, −15.56256155725142146467256339543, −14.71397054292797476282937160186, −13.99554760260377734316242332386, −12.48159019124484424568499189773, −11.9370906606568005826718246379, −11.23019130470583588336979212364, −10.26792391708813578708470242952, −9.62178851427619966459264056048, −8.42020441672029516555208800105, −7.616405386147492929074351916883, −6.97454900514665151085279546572, −6.065988965977424411414221906734, −5.54274379053284101005043575455, −4.36225087298857701152799123953, −2.86532892701944110181757630384, −1.78777848479410762401955121482, −0.42487032861929982758135589411, 0.845150560389679104494619708319, 1.621621892225413249660574424138, 3.55572950399945645887604184206, 3.97521810876726788028076553345, 4.799455885480161214056128380253, 6.196630164992983200540569812253, 7.12679529240544193446455565653, 7.849762077588452280557983520571, 9.011002406444105072174267754394, 9.65979448958198741492824146458, 10.40397003340846656749449922878, 11.32483261280425569757963153260, 11.94295775606473371560597831795, 12.4326772799018087633895756871, 13.49713296149784365268358191414, 14.652921280233060622873228318316, 15.88593010157764050839880876819, 16.47459047283082422399869113112, 16.98617515244021033417428382889, 17.45046168248134099596889724091, 18.643506351668771314103283643554, 19.49737724657798034717555050059, 19.92486696323352434467476604567, 20.92498449457266998793500859528, 21.42359041939138965236112418571

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