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Properties

Label 1-6048-6048.2459-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.523 + 0.852i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
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  + (0.906 + 0.422i)5-s + (0.422 + 0.906i)11-s + (−0.422 + 0.906i)13-s + (−0.5 − 0.866i)17-s + (−0.258 − 0.965i)19-s + (−0.984 − 0.173i)23-s + (0.642 + 0.766i)25-s + (−0.906 + 0.422i)29-s + (0.939 − 0.342i)31-s + (0.707 + 0.707i)37-s + (−0.342 − 0.939i)41-s + (−0.573 + 0.819i)43-s + (0.939 + 0.342i)47-s + (−0.965 + 0.258i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.906 + 0.422i)5-s + (0.422 + 0.906i)11-s + (−0.422 + 0.906i)13-s + (−0.5 − 0.866i)17-s + (−0.258 − 0.965i)19-s + (−0.984 − 0.173i)23-s + (0.642 + 0.766i)25-s + (−0.906 + 0.422i)29-s + (0.939 − 0.342i)31-s + (0.707 + 0.707i)37-s + (−0.342 − 0.939i)41-s + (−0.573 + 0.819i)43-s + (0.939 + 0.342i)47-s + (−0.965 + 0.258i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.523 + 0.852i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.523 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7043692181 + 1.258973250i\)
\(L(\frac12)\) \(\approx\) \(0.7043692181 + 1.258973250i\)
\(L(1)\) \(\approx\) \(1.086078170 + 0.2660877515i\)
\(L(1)\) \(\approx\) \(1.086078170 + 0.2660877515i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.906 + 0.422i)T \)
11 \( 1 + (0.422 + 0.906i)T \)
13 \( 1 + (-0.422 + 0.906i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (-0.906 + 0.422i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (-0.573 + 0.819i)T \)
47 \( 1 + (0.939 + 0.342i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (0.0871 + 0.996i)T \)
61 \( 1 + (0.906 - 0.422i)T \)
67 \( 1 + (-0.573 - 0.819i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.906 + 0.422i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−17.313718753505735124012978063551, −17.02194155483039525157570302627, −16.23651582083510032336360176448, −15.61103612303608685415374790710, −14.65485199160194491972024346028, −14.31075127795092426230700371284, −13.391982164621958737491684390237, −13.04257586395443002961769355943, −12.28967534653806091230992467066, −11.595159801292087284640373064818, −10.74059133890238711638144120710, −10.08890275054710881418867364048, −9.66911191807589370169992898582, −8.60425196260833196000249642498, −8.3679286895763804414180302742, −7.48325919590976341210278240922, −6.43550006408552740355290877708, −5.89718278075065992163569504087, −5.50228530007819739080889924019, −4.46770405302482629712181261960, −3.78403122462864117916018122195, −2.91924608208245867213316775509, −2.037435387911529680447480215324, −1.40852568084033775382837009927, −0.34263423547110277913598744326, 1.135239870719139345688940411, 2.154263961115535753196765107413, 2.37584068064528534780639039033, 3.43129548598259033455440071118, 4.53577757710281160849966776607, 4.77611389017560956406029448871, 5.8753162770606867029914089415, 6.5456358902444835576623275315, 7.02718436850631121226362173105, 7.70973384268431053947679535431, 8.85035265497468270616221354305, 9.37848219238070589620836046444, 9.81919894083828281748503417358, 10.588944099205752760017529874477, 11.37956841369909864355910195954, 11.92564239062288680651278442608, 12.70616740839209598089259445889, 13.53386343691031466556376106359, 13.88990898421234383586737431907, 14.67935466044423937335731602644, 15.15496796560264068563199499348, 15.97449413896666632199252613564, 16.7795484206394892967824927266, 17.3142348718588179608746719460, 17.87636253649371152603264068353

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