↑

Properties

Label 1-1573-1573.1200-r0-0-0
Degree $1$
Conductor $1573$
Sign $0.758 + 0.651i$
Analytic cond. $7.30498$
Root an. cond. $7.30498$
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  + (−0.580 + 0.814i)2-s + (−0.5 − 0.866i)3-s + (−0.327 − 0.945i)4-s + (0.959 − 0.281i)5-s + (0.995 + 0.0950i)6-s + (0.786 + 0.618i)7-s + (0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.327 + 0.945i)10-s + (−0.654 + 0.755i)12-s + (−0.959 + 0.281i)14-s + (−0.723 − 0.690i)15-s + (−0.786 + 0.618i)16-s + (0.0475 + 0.998i)17-s + (−0.415 − 0.909i)18-s + (0.888 − 0.458i)19-s + ⋯
L(s)  = 1  + (−0.580 + 0.814i)2-s + (−0.5 − 0.866i)3-s + (−0.327 − 0.945i)4-s + (0.959 − 0.281i)5-s + (0.995 + 0.0950i)6-s + (0.786 + 0.618i)7-s + (0.959 + 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.327 + 0.945i)10-s + (−0.654 + 0.755i)12-s + (−0.959 + 0.281i)14-s + (−0.723 − 0.690i)15-s + (−0.786 + 0.618i)16-s + (0.0475 + 0.998i)17-s + (−0.415 − 0.909i)18-s + (0.888 − 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1573\)    =    \(11^{2} \cdot 13\)
Sign: $0.758 + 0.651i$
Analytic conductor: \(7.30498\)
Root analytic conductor: \(7.30498\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1573} (1200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1573,\ (0:\ ),\ 0.758 + 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.175243951 + 0.4353434653i\)
\(L(\frac12)\) \(\approx\) \(1.175243951 + 0.4353434653i\)
\(L(1)\) \(\approx\) \(0.8786900717 + 0.1481616617i\)
\(L(1)\) \(\approx\) \(0.8786900717 + 0.1481616617i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.580 + 0.814i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.786 + 0.618i)T \)
17 \( 1 + (0.0475 + 0.998i)T \)
19 \( 1 + (0.888 - 0.458i)T \)
23 \( 1 + (-0.786 + 0.618i)T \)
29 \( 1 + (0.0475 - 0.998i)T \)
31 \( 1 + (0.654 + 0.755i)T \)
37 \( 1 + (0.327 - 0.945i)T \)
41 \( 1 + (-0.580 + 0.814i)T \)
43 \( 1 + (0.723 - 0.690i)T \)
47 \( 1 + (-0.415 + 0.909i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (-0.580 - 0.814i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
67 \( 1 + (-0.580 + 0.814i)T \)
71 \( 1 + (0.888 - 0.458i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (0.888 + 0.458i)T \)
97 \( 1 + (-0.723 + 0.690i)T \)
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

−20.543469605251084905624388486379, −20.06764149765769046459948585911, −18.70177071213361172042238948164, −18.00159982911728192117795488717, −17.70042534210623696966952235501, −16.66432676766196047435922276109, −16.46426791803803177769208697600, −15.22625018083780745936324495986, −14.095594437588039702202572088435, −13.87204778802273367279761559641, −12.66728955301406854625318428622, −11.706803745632613256613987045203, −11.2440438442149701013424642415, −10.30465026362415476872450146986, −10.00183284367244867808830250942, −9.2251065043449948868147858632, −8.34020514822029619052435619773, −7.35626914756769866611326164523, −6.40665721652631464386420964708, −5.23848753716004039078595113681, −4.66830776869094708051714930555, −3.65102961169246122702261143853, −2.811166091339427737292582477006, −1.73873074583271521579532862821, −0.73377733285860086402786974233, 1.04712713807929879384617585294, 1.70290639259723560949190282400, 2.531799448864702927591462308846, 4.46956630100659262772803095115, 5.34349281121540745536170027146, 5.8484346133872577114786306823, 6.47203038238097019182416103372, 7.52451879316520688314193858648, 8.150086903604200137433523569688, 8.894137454823859597231091675823, 9.75245252088317216237532062676, 10.6160313246502574480268957529, 11.43796261929917231520007782656, 12.30962369078556583230547831229, 13.228107590774689248260103756029, 13.929663141921435075745066856668, 14.46878947799111635340374928013, 15.51577749096501798555765279287, 16.2503765227485971344284415627, 17.28173277774898224654842320151, 17.52954753604774436680679675209, 18.09642648854792306091777655195, 18.82149003338354697786588189667, 19.57769062825613401380778476761, 20.43404520964211042014298641442

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