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Properties

Label 1-4896-4896.1645-r0-0-0
Degree $1$
Conductor $4896$
Sign $0.969 - 0.245i$
Analytic cond. $22.7369$
Root an. cond. $22.7369$
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  + (−0.965 + 0.258i)5-s + (0.5 + 0.866i)7-s + (0.258 − 0.965i)11-s + (−0.965 + 0.258i)13-s + (−0.707 − 0.707i)19-s + (−0.5 + 0.866i)23-s + (0.866 − 0.5i)25-s + (−0.965 − 0.258i)29-s + (−0.866 − 0.5i)31-s + (−0.707 − 0.707i)35-s + (0.707 + 0.707i)37-s + (−0.5 + 0.866i)41-s + (−0.965 − 0.258i)43-s + (0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)5-s + (0.5 + 0.866i)7-s + (0.258 − 0.965i)11-s + (−0.965 + 0.258i)13-s + (−0.707 − 0.707i)19-s + (−0.5 + 0.866i)23-s + (0.866 − 0.5i)25-s + (−0.965 − 0.258i)29-s + (−0.866 − 0.5i)31-s + (−0.707 − 0.707i)35-s + (0.707 + 0.707i)37-s + (−0.5 + 0.866i)41-s + (−0.965 − 0.258i)43-s + (0.5 + 0.866i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4896\)    =    \(2^{5} \cdot 3^{2} \cdot 17\)
Sign: $0.969 - 0.245i$
Analytic conductor: \(22.7369\)
Root analytic conductor: \(22.7369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4896} (1645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4896,\ (0:\ ),\ 0.969 - 0.245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9704128310 - 0.1211454648i\)
\(L(\frac12)\) \(\approx\) \(0.9704128310 - 0.1211454648i\)
\(L(1)\) \(\approx\) \(0.8191444851 + 0.05613608235i\)
\(L(1)\) \(\approx\) \(0.8191444851 + 0.05613608235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.965 - 0.258i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.258 - 0.965i)T \)
61 \( 1 + (0.965 + 0.258i)T \)
67 \( 1 + (0.965 - 0.258i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (-0.258 + 0.965i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−18.17676190583331802855937944033, −17.25458200998854671141574713128, −16.82997390640669036782199317185, −16.27249268091596711840727442466, −15.31890308438075459996054987030, −14.648668381777017827054058998285, −14.49056427531293511820192176823, −13.32356594648659119798322496459, −12.584338920387104178659022088354, −12.17798034949478204349740593524, −11.43931006342858326327762474237, −10.59376683447344243640132010247, −10.18941753584658713956949409833, −9.24482986355695115080834167385, −8.455404760861806041815637631902, −7.75040890888051792742443903394, −7.25212720467033154080733901425, −6.686482860109126947180574830458, −5.430926187645676489156185268779, −4.81211992360197554830654879389, −4.01910358672054080242024362127, −3.730208225195345456791676492564, −2.40157528125858834325778029270, −1.69519160146297970999248911351, −0.601922346963224112187778534311, 0.417099904230326324892191393395, 1.72316781544405438426644311602, 2.49856671576993333193421886955, 3.30002313361182258538669576002, 4.0466078191884967237094286879, 4.84472950505727832993072575889, 5.542189903517542075429756123631, 6.37704804584101255429019738459, 7.1297472019491598590215053285, 7.93712095396287803890081366321, 8.37992836180627578918011814345, 9.1848481408394568320883150353, 9.82089571561266876290822132291, 10.94524019286807655251746720292, 11.4408402823725072262601523612, 11.77137087831058732687878134336, 12.63515975968339559788958156888, 13.318243238358686038308773131176, 14.26380966493810269558539611953, 14.87342488519012764600843679438, 15.25925950505440172761798199257, 16.00870894454112063014644756053, 16.74216973659533505455902982194, 17.309402793589027131621495261541, 18.2792909030392287527970715630

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