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Properties

Label 1-6048-6048.2707-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.849 + 0.527i$
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  + (−0.996 + 0.0871i)5-s + (0.0871 − 0.996i)11-s + (−0.0871 − 0.996i)13-s + (−0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.996 − 0.0871i)29-s + (0.766 + 0.642i)31-s + (0.707 − 0.707i)37-s + (0.642 − 0.766i)41-s + (−0.906 + 0.422i)43-s + (−0.766 + 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.996 + 0.0871i)5-s + (0.0871 − 0.996i)11-s + (−0.0871 − 0.996i)13-s + (−0.5 − 0.866i)17-s + (−0.965 + 0.258i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.996 − 0.0871i)29-s + (0.766 + 0.642i)31-s + (0.707 − 0.707i)37-s + (0.642 − 0.766i)41-s + (−0.906 + 0.422i)43-s + (−0.766 + 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.849 + 0.527i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.849 + 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05029149881 - 0.1762834251i\)
\(L(\frac12)\) \(\approx\) \(-0.05029149881 - 0.1762834251i\)
\(L(1)\) \(\approx\) \(0.7250481303 - 0.1372478847i\)
\(L(1)\) \(\approx\) \(0.7250481303 - 0.1372478847i\)

Euler product

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

−18.23512093277728428718665946793, −17.1533320000176716429989949935, −16.908924517141488754717080646214, −16.141053443448677640637745394194, −15.30954640986392795099681873811, −14.89442824426442611846715992079, −14.46774877210143517745566121677, −13.18312593728779186785478156883, −12.954805796616466092727155668696, −12.05488526354133341661400796570, −11.58800738290615847300626558530, −10.89305033843833760707570270963, −10.19704929578526978806363427962, −9.37711767440126216940448944410, −8.66439892339187434703717879978, −8.137274168595504050571273989129, −7.28659640036007386833031877395, −6.72444249137838806056585383632, −6.116133576027850700076510785, −4.88978772703815698028099329514, −4.30805473248599287836385649980, −4.01987341294702331234266153381, −2.84923729766626503864057876564, −2.106756947868889833293933402556, −1.26008994584816836717753592778, 0.05917625185824564754812346773, 0.82730651437051783564152768754, 1.97042349343146800860915286428, 3.08448434709980229955849526781, 3.35307873810204684213935630413, 4.30636454499174693764992916718, 4.99190913306454711448359103262, 5.81040648675059035429374037746, 6.52316929545958743972544364818, 7.38745869872934847941262541532, 7.89584597489053510596078719812, 8.57238628598041887175945971379, 9.19401887821258081979042630927, 10.096288617084301128772885402291, 10.94514682877810694648689723232, 11.26106650534768550611111831123, 11.95007773198762779743817880373, 12.80756116069188309245238105173, 13.24052293844632336147247441153, 14.11006764140623582717970289419, 14.8333553451009278161027703357, 15.348459869829135092579817565133, 16.07269511581219398959543871253, 16.44422996961190812774517728799, 17.38457036071257469663114896104

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