↑

Properties

Label 1-703-703.256-r0-0-0
Degree $1$
Conductor $703$
Sign $0.569 - 0.822i$
Analytic cond. $3.26471$
Root an. cond. $3.26471$
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  + (−0.5 − 0.866i)2-s + (−0.939 − 0.342i)3-s + (−0.5 + 0.866i)4-s + (0.173 + 0.984i)5-s + (0.173 + 0.984i)6-s + (0.173 − 0.984i)7-s + 8-s + (0.766 + 0.642i)9-s + (0.766 − 0.642i)10-s + 11-s + (0.766 − 0.642i)12-s + (−0.5 − 0.866i)13-s + (−0.939 + 0.342i)14-s + (0.173 − 0.984i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.939 − 0.342i)3-s + (−0.5 + 0.866i)4-s + (0.173 + 0.984i)5-s + (0.173 + 0.984i)6-s + (0.173 − 0.984i)7-s + 8-s + (0.766 + 0.642i)9-s + (0.766 − 0.642i)10-s + 11-s + (0.766 − 0.642i)12-s + (−0.5 − 0.866i)13-s + (−0.939 + 0.342i)14-s + (0.173 − 0.984i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 703 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 703 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(703\)    =    \(19 \cdot 37\)
Sign: $0.569 - 0.822i$
Analytic conductor: \(3.26471\)
Root analytic conductor: \(3.26471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{703} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 703,\ (0:\ ),\ 0.569 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7438882806 - 0.3898162307i\)
\(L(\frac12)\) \(\approx\) \(0.7438882806 - 0.3898162307i\)
\(L(1)\) \(\approx\) \(0.6551032891 - 0.2570436981i\)
\(L(1)\) \(\approx\) \(0.6551032891 - 0.2570436981i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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

−22.88271260230189654094074465532, −22.012466365164512769195398600270, −21.360658261992964350703001945683, −20.34525693183250569836450228027, −19.115212410846523868708472602838, −18.62516640864947802715360637255, −17.39548502571933704258133217185, −17.12595142696747264308011631828, −16.29179776054769662464670989155, −15.71004529158914864768513005945, −14.698516929062921844725468294169, −13.96118938326225743967002151307, −12.43959286748490232672761689683, −12.13048447074904635461461649012, −10.993276547235904717065985845014, −9.810661513108570691566741944, −9.235477671266370706089976918839, −8.56716179004455157365544841790, −7.3233464878892898736978014343, −6.29131966467814731847759780070, −5.65111632671712975862481154769, −4.838981161523105617702904280611, −4.091923541218159963474976461785, −1.908702736523557588000290136390, −0.8552201022335725698034102405, 0.84801473798581975700766165520, 1.76702480275242934634710708699, 3.14284959502130256680536205307, 3.97587581448790536665967028972, 5.16287002657573214679046438007, 6.341239469830091043989314103099, 7.42754746734756066255754420357, 7.697176055080115970503178210334, 9.41769691182612841051305823669, 10.182074827640559582495278893917, 10.79057986655742196502785013773, 11.49991876384383641801660200745, 12.27121626618598002092769642083, 13.234066701929404268440321894859, 14.0147656185881922920057206030, 14.98520040731696619203446315759, 16.50809263624198154133594973371, 16.9867225540444812069871946520, 17.81337772731929794064833483405, 18.251165240506875753584483449262, 19.38615697154280131005632295099, 19.73473492724025349548178904622, 20.96442721062337590885179580209, 21.785733132234441298148690245947, 22.50859540007148211977735742787

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