↑

Properties

Label 1-2432-2432.2051-r0-0-0
Degree $1$
Conductor $2432$
Sign $0.857 + 0.514i$
Analytic cond. $11.2941$
Root an. cond. $11.2941$
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  + (−0.195 + 0.980i)3-s + (0.831 + 0.555i)5-s + (−0.923 + 0.382i)7-s + (−0.923 − 0.382i)9-s + (−0.980 + 0.195i)11-s + (0.831 − 0.555i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.195 − 0.980i)21-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (0.555 − 0.831i)27-s + (0.980 + 0.195i)29-s − i·31-s − i·33-s + ⋯
L(s)  = 1  + (−0.195 + 0.980i)3-s + (0.831 + 0.555i)5-s + (−0.923 + 0.382i)7-s + (−0.923 − 0.382i)9-s + (−0.980 + 0.195i)11-s + (0.831 − 0.555i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.195 − 0.980i)21-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (0.555 − 0.831i)27-s + (0.980 + 0.195i)29-s − i·31-s − i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2432\)    =    \(2^{7} \cdot 19\)
Sign: $0.857 + 0.514i$
Analytic conductor: \(11.2941\)
Root analytic conductor: \(11.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2432} (2051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2432,\ (0:\ ),\ 0.857 + 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262171132 + 0.3492897936i\)
\(L(\frac12)\) \(\approx\) \(1.262171132 + 0.3492897936i\)
\(L(1)\) \(\approx\) \(0.9237768143 + 0.3080301719i\)
\(L(1)\) \(\approx\) \(0.9237768143 + 0.3080301719i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.195 + 0.980i)T \)
5 \( 1 + (0.831 + 0.555i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (-0.980 + 0.195i)T \)
13 \( 1 + (0.831 - 0.555i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.980 + 0.195i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.555 - 0.831i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (0.195 + 0.980i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.980 - 0.195i)T \)
59 \( 1 + (-0.831 - 0.555i)T \)
61 \( 1 + (0.195 - 0.980i)T \)
67 \( 1 + (0.195 - 0.980i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.555 + 0.831i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
97 \( 1 - iT \)
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

−19.401460862583980696286512696252, −18.73567654739051709508024127208, −17.95639148194239536870867067233, −17.44202977030259014112949144430, −16.67291938944366036710661807176, −16.07921407263028509673549384401, −15.28235361464288110514750542201, −13.89340881159282038953563157107, −13.65259765403919922724480198147, −13.04493966947897079343371901517, −12.5015441889648526437034519271, −11.59782926184495316294520530911, −10.63561869642677997490353692718, −10.13549099797736096925195555195, −8.94275830609190981882746743701, −8.61051055348806090269270540987, −7.560619118747452056524845995630, −6.71792821392384187368073628919, −6.1520626503631317430994476053, −5.5085506408881590734881262137, −4.558922849927197105660049050506, −3.37575977037975513555498047381, −2.52817544894723196363920201203, −1.63781353918783481671000304110, −0.82241286125356048265551984954, 0.56278602509476316112263205002, 2.32248835389936848886617641364, 2.82901201508939187536332088475, 3.56334595576433735607268091514, 4.72155475281774772504824484187, 5.3723533379362209945069276261, 6.23401355096509720966255300204, 6.6143285608643343920214777896, 7.92295352715055145260989498294, 8.820997737781184970279439885946, 9.53536315918696491802401870665, 10.07352693060862765909708305356, 10.772007054114444892341116184897, 11.29576814587871893875591520385, 12.4646240403008040869991048524, 13.2082582627211322117682045114, 13.75286601534222977367118400460, 14.81396681772701598865107720063, 15.28649380531860012386204524490, 16.07461821883483164194495585387, 16.4820284055477714641517146035, 17.512987186052610899485782667158, 18.199347095480393369977296271714, 18.57289372991913238952751881085, 19.75093106623826285644426662470

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