↑

Properties

Label 1-592-592.11-r1-0-0
Degree $1$
Conductor $592$
Sign $-0.995 + 0.0983i$
Analytic cond. $63.6192$
Root an. cond. $63.6192$
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  + (0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s − i·11-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)15-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)21-s − 23-s + (0.5 − 0.866i)25-s − i·27-s − i·29-s + 31-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s − i·11-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)15-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.866 − 0.5i)21-s − 23-s + (0.5 − 0.866i)25-s − i·27-s − i·29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(592\)    =    \(2^{4} \cdot 37\)
Sign: $-0.995 + 0.0983i$
Analytic conductor: \(63.6192\)
Root analytic conductor: \(63.6192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{592} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 592,\ (1:\ ),\ -0.995 + 0.0983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05242342673 - 1.063630616i\)
\(L(\frac12)\) \(\approx\) \(-0.05242342673 - 1.063630616i\)
\(L(1)\) \(\approx\) \(0.9622267471 - 0.4147930406i\)
\(L(1)\) \(\approx\) \(0.9622267471 - 0.4147930406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 - iT \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 - T \)
29 \( 1 - iT \)
31 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 - T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 - T \)
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

−23.35718204551768696675885316834, −22.51220265042779376823187676629, −21.50797306384138558979910575425, −20.86641184075608305083697114300, −20.04295119187818996344763463232, −19.213498138817047277998760577358, −18.81962867281616189410118716930, −17.45001550745401171086300810447, −16.39006148666551270860555685324, −15.54244217318294211319979223399, −15.29573568228890283831408509897, −14.23622999531032434275813236399, −13.10562237787067368733350466350, −12.43698327802351524978044457975, −11.53011691335306538051883586978, −10.32836083383026757615133282649, −9.48230743101660754561189212188, −8.58442410022970983216316679784, −8.10220938513976525389101716899, −6.89846216864187097594776545800, −5.699422926560594306983633036456, −4.375789730328882447295196785567, −3.92628823430138325601537786730, −2.68617881480490406902163085605, −1.65229122662054948762026293035, 0.23386686252963167278383110135, 1.27624250397461564553956787614, 2.97883959480492245212899219161, 3.41309608030641286491769184790, 4.355798367249916367000711544051, 6.15314844938299878421552345054, 6.80320582722265332435594399230, 7.98036901034458457686965695642, 8.21285560081584620328000564577, 9.55576962113568971276831441556, 10.52148148707785862856773766345, 11.376049842379989406013479800229, 12.44858599311519669194976312028, 13.30105596819043899085946703598, 14.0862922288505788499198611377, 14.7166458744394105902144728933, 15.9591144425435604510850623634, 16.273466448493526400712170141108, 17.74911354502297904450921055125, 18.63131575174350294320340637873, 19.17891730615722097134705343385, 19.95550359961325285070037774257, 20.59234782227543914906066467073, 21.57449395391949304142650163946, 22.8292759047299203648486079575

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