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Properties

Label 1-592-592.323-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

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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: $\chi_{592} (323, \cdot )$
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 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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