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Properties

Label 1-6048-6048.3397-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.850 - 0.526i$
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  + (−0.422 − 0.906i)5-s + (0.906 + 0.422i)11-s + (0.819 + 0.573i)13-s − 17-s + (0.707 − 0.707i)19-s + (0.984 − 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.573 + 0.819i)29-s + (0.766 − 0.642i)31-s + (−0.965 + 0.258i)37-s + (0.984 − 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.766 − 0.642i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.422 − 0.906i)5-s + (0.906 + 0.422i)11-s + (0.819 + 0.573i)13-s − 17-s + (0.707 − 0.707i)19-s + (0.984 − 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.573 + 0.819i)29-s + (0.766 − 0.642i)31-s + (−0.965 + 0.258i)37-s + (0.984 − 0.173i)41-s + (−0.0871 − 0.996i)43-s + (−0.766 − 0.642i)47-s + (0.965 − 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.850 - 0.526i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (3397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.850 - 0.526i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.849729275 - 0.5266568164i\)
\(L(\frac12)\) \(\approx\) \(1.849729275 - 0.5266568164i\)
\(L(1)\) \(\approx\) \(1.126343382 - 0.1565358807i\)
\(L(1)\) \(\approx\) \(1.126343382 - 0.1565358807i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.422 - 0.906i)T \)
11 \( 1 + (0.906 + 0.422i)T \)
13 \( 1 + (0.819 + 0.573i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.573 + 0.819i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (-0.0871 - 0.996i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
59 \( 1 + (-0.573 + 0.819i)T \)
61 \( 1 + (-0.996 + 0.0871i)T \)
67 \( 1 + (0.906 - 0.422i)T \)
71 \( 1 + (0.866 - 0.5i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.573 - 0.819i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−17.77375073572235249139122985456, −17.26450758235069928261188732743, −16.310208225369283985278547439985, −15.69452635359486568766311616432, −15.27735691796338251092540951992, −14.3737685233129473435174568914, −13.98613774690388576948571849774, −13.26582907698895570298231789880, −12.43410195888529268952219819207, −11.65802309166523382709611694522, −11.17565112479110834736000915097, −10.62946577946905703056238829298, −9.84420269707718814561029607635, −9.075206584215544389345691186455, −8.338835729711893784002710744138, −7.75425547873938257567094017562, −6.85733626616277447478647305191, −6.38719375585445322516055005874, −5.74678537501504375936944252758, −4.69792344969284610325710329402, −3.94599752415582789184023433287, −3.267715089335186096855652539937, −2.73428359773218631265444901730, −1.598757551842527750108026549580, −0.77130287016911235098517531874, 0.708016623766501089678093328933, 1.38608110863600769294839597004, 2.236888870882213582921177897586, 3.32788956861334471869480986581, 4.02062245281801530411641512847, 4.6809601403252251131633057266, 5.21434160134108486297999357320, 6.29230885414754589754440087715, 6.85268329874027284267259734487, 7.52328192848382493998248960482, 8.56789875141132809235119648992, 8.92785790227401987422280740463, 9.38603731714171127566547002775, 10.41101652941873203535819228556, 11.21104310162551425116564085339, 11.74238524919432943611667441196, 12.28375912043785076292559675455, 13.13785406415160834928745736774, 13.593760245938646270089268410171, 14.30259547522408358141517703059, 15.30151716968319655634936571036, 15.59209379877787500185264473250, 16.38316073789732226492249277516, 16.95990281032559223432617167433, 17.53572759688438159759227768055

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