↑

Properties

Label 1-1007-1007.486-r0-0-0
Degree $1$
Conductor $1007$
Sign $0.565 + 0.824i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.428 + 0.903i)2-s + (0.200 − 0.979i)3-s + (−0.632 − 0.774i)4-s + (−0.799 + 0.600i)5-s + (0.799 + 0.600i)6-s + (0.568 + 0.822i)7-s + (0.970 − 0.239i)8-s + (−0.919 − 0.391i)9-s + (−0.200 − 0.979i)10-s + (0.885 − 0.464i)11-s + (−0.885 + 0.464i)12-s + (0.987 + 0.160i)13-s + (−0.987 + 0.160i)14-s + (0.428 + 0.903i)15-s + (−0.200 + 0.979i)16-s + (0.278 + 0.960i)17-s + ⋯
L(s)  = 1  + (−0.428 + 0.903i)2-s + (0.200 − 0.979i)3-s + (−0.632 − 0.774i)4-s + (−0.799 + 0.600i)5-s + (0.799 + 0.600i)6-s + (0.568 + 0.822i)7-s + (0.970 − 0.239i)8-s + (−0.919 − 0.391i)9-s + (−0.200 − 0.979i)10-s + (0.885 − 0.464i)11-s + (−0.885 + 0.464i)12-s + (0.987 + 0.160i)13-s + (−0.987 + 0.160i)14-s + (0.428 + 0.903i)15-s + (−0.200 + 0.979i)16-s + (0.278 + 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1007 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1007\)    =    \(19 \cdot 53\)
Sign: $0.565 + 0.824i$
Analytic conductor: \(4.67648\)
Root analytic conductor: \(4.67648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1007} (486, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1007,\ (0:\ ),\ 0.565 + 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.000110219 + 0.5267811385i\)
\(L(\frac12)\) \(\approx\) \(1.000110219 + 0.5267811385i\)
\(L(1)\) \(\approx\) \(0.8469486179 + 0.2331380036i\)
\(L(1)\) \(\approx\) \(0.8469486179 + 0.2331380036i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.428 + 0.903i)T \)
3 \( 1 + (0.200 - 0.979i)T \)
5 \( 1 + (-0.799 + 0.600i)T \)
7 \( 1 + (0.568 + 0.822i)T \)
11 \( 1 + (0.885 - 0.464i)T \)
13 \( 1 + (0.987 + 0.160i)T \)
17 \( 1 + (0.278 + 0.960i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.845 + 0.534i)T \)
31 \( 1 + (-0.885 - 0.464i)T \)
37 \( 1 + (-0.748 - 0.663i)T \)
41 \( 1 + (0.845 + 0.534i)T \)
43 \( 1 + (0.948 + 0.316i)T \)
47 \( 1 + (0.799 + 0.600i)T \)
59 \( 1 + (-0.919 + 0.391i)T \)
61 \( 1 + (-0.278 + 0.960i)T \)
67 \( 1 + (0.632 + 0.774i)T \)
71 \( 1 + (-0.948 - 0.316i)T \)
73 \( 1 + (-0.278 - 0.960i)T \)
79 \( 1 + (0.996 + 0.0804i)T \)
83 \( 1 - T \)
89 \( 1 + (0.692 - 0.721i)T \)
97 \( 1 + (0.799 - 0.600i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.115295423191176051307814961284, −20.61738912522792024253832851137, −20.16981525962900447868920752456, −19.50065170739762653993508186943, −18.56841525399036954760199713338, −17.38636440314318770311176112586, −16.974743167887518454155786639090, −16.138334603569379893252220362344, −15.36588454634049344480773630178, −14.240770297302289890659821346636, −13.63890842548607820700669851955, −12.55913108401415461574922655377, −11.53227969465017819553037655815, −11.21001603965669148215352571069, −10.34880710428601210931978298041, −9.29186910346837220167916954496, −8.91885194571894149830309983218, −7.89391854949605393397068390640, −7.2421387994239396603387335420, −5.3782158746046366764103807972, −4.55871276064967119283130276131, −3.82722922715323828559554350155, −3.35545349718958823156921490998, −1.78212454024944692256575925882, −0.737830821237727559446812171404, 0.98886136173271611371178873586, 1.96622872094919921671419508530, 3.34786314694833052573551501363, 4.31824836601554145851345733764, 5.85561596173801029170790563635, 6.144097927080014309752910815527, 7.21126131888313492632451913285, 7.829721523282812075019648445520, 8.78444512794240093690205479592, 8.97413527435360667632515101752, 10.76339520613105473690958584986, 11.23936952490060016674606459395, 12.26852963580294360078315143514, 13.09569630869959510889463677144, 14.37908791979795987545188867907, 14.480982174463808141376520542312, 15.351392524605029622887864048866, 16.29927641989919626108052279828, 17.123243276236748124486020126875, 18.056997653430392957294581083684, 18.574077776185697304654838957464, 19.13404311850147625412262827645, 19.73326412422332542606700054747, 20.86070607738897548098572343559, 22.13422828674785960337052821758

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