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Properties

Label 1-4608-4608.1051-r1-0-0
Degree $1$
Conductor $4608$
Sign $-0.706 - 0.708i$
Analytic cond. $495.198$
Root an. cond. $495.198$
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  + (0.569 + 0.822i)5-s + (0.683 − 0.729i)7-s + (0.783 − 0.621i)11-s + (−0.762 + 0.646i)13-s + (0.980 − 0.195i)17-s + (−0.671 + 0.740i)19-s + (−0.412 + 0.910i)23-s + (−0.352 + 0.935i)25-s + (−0.917 − 0.397i)29-s + (−0.793 + 0.608i)31-s + (0.989 + 0.146i)35-s + (−0.998 − 0.0490i)37-s + (−0.352 − 0.935i)41-s + (0.274 + 0.961i)43-s + (−0.0654 + 0.997i)47-s + ⋯
L(s)  = 1  + (0.569 + 0.822i)5-s + (0.683 − 0.729i)7-s + (0.783 − 0.621i)11-s + (−0.762 + 0.646i)13-s + (0.980 − 0.195i)17-s + (−0.671 + 0.740i)19-s + (−0.412 + 0.910i)23-s + (−0.352 + 0.935i)25-s + (−0.917 − 0.397i)29-s + (−0.793 + 0.608i)31-s + (0.989 + 0.146i)35-s + (−0.998 − 0.0490i)37-s + (−0.352 − 0.935i)41-s + (0.274 + 0.961i)43-s + (−0.0654 + 0.997i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.706 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-0.706 - 0.708i$
Analytic conductor: \(495.198\)
Root analytic conductor: \(495.198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4608,\ (1:\ ),\ -0.706 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1229640892 - 0.2962900640i\)
\(L(\frac12)\) \(\approx\) \(0.1229640892 - 0.2962900640i\)
\(L(1)\) \(\approx\) \(1.100043497 + 0.1203561426i\)
\(L(1)\) \(\approx\) \(1.100043497 + 0.1203561426i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.569 + 0.822i)T \)
7 \( 1 + (0.683 - 0.729i)T \)
11 \( 1 + (0.783 - 0.621i)T \)
13 \( 1 + (-0.762 + 0.646i)T \)
17 \( 1 + (0.980 - 0.195i)T \)
19 \( 1 + (-0.671 + 0.740i)T \)
23 \( 1 + (-0.412 + 0.910i)T \)
29 \( 1 + (-0.917 - 0.397i)T \)
31 \( 1 + (-0.793 + 0.608i)T \)
37 \( 1 + (-0.998 - 0.0490i)T \)
41 \( 1 + (-0.352 - 0.935i)T \)
43 \( 1 + (0.274 + 0.961i)T \)
47 \( 1 + (-0.0654 + 0.997i)T \)
53 \( 1 + (0.803 + 0.595i)T \)
59 \( 1 + (-0.179 + 0.983i)T \)
61 \( 1 + (0.0163 - 0.999i)T \)
67 \( 1 + (0.485 + 0.874i)T \)
71 \( 1 + (-0.471 + 0.881i)T \)
73 \( 1 + (-0.290 + 0.956i)T \)
79 \( 1 + (0.442 - 0.896i)T \)
83 \( 1 + (-0.541 - 0.840i)T \)
89 \( 1 + (-0.995 - 0.0980i)T \)
97 \( 1 + (0.991 - 0.130i)T \)
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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

−18.15173251134635919588873969086, −17.602201954433979075496652659098, −16.88069555317168584590807965013, −16.60919857378459176923393590725, −15.43759243487830802859150322211, −14.87300237789692959765403610250, −14.449412040921865675781415916605, −13.54198893511703101978812475685, −12.69841591757224694938618289658, −12.309405474164889485588223326517, −11.74079176041232484735225288596, −10.75627636184344811228920693892, −10.01303162543220601624232494347, −9.35729695354064939119628751743, −8.7322274799782876822505313141, −8.123294709211763528637982580304, −7.29301336796874644636192603100, −6.431231464953201595593875784360, −5.54853417058007174841750423662, −5.12133501015520939905933983310, −4.41208873661873361821162135539, −3.50113816232139439253972414490, −2.254701241159939180249343544194, −1.94700414676316580141821360173, −0.95207307906726435302289423462, 0.04292962390274909993047298462, 1.422946832085941941238774090676, 1.75466835933418836794072094892, 2.87047251153054015112043053085, 3.71922569939623727737140621322, 4.22784967917552430081995535188, 5.42569650336005276936785474385, 5.84024192358646302707595184030, 6.86562572865304312161228067391, 7.29944381591400709183730569340, 8.02416392089668308758940449467, 8.98705450641948159394470502714, 9.68641670906774698730130287921, 10.30669271491426462712235060584, 10.98172250527657852375055655347, 11.60842093754298544522846720452, 12.24671749822160843700035487493, 13.2596029433497052091928170973, 13.99690586494204388164216641111, 14.43336384813225970040202211947, 14.70764742231008199107590393216, 15.822129494180182058354320263208, 16.71017658654731449394771676178, 17.1249401672303933886110835279, 17.638505452293661870302677925080

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