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Properties

Label 1-6048-6048.2291-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.411 + 0.911i$
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  + (−0.819 − 0.573i)5-s + (0.573 + 0.819i)11-s + (0.996 + 0.0871i)13-s + 17-s + (−0.707 − 0.707i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (0.0871 + 0.996i)29-s + (0.939 + 0.342i)31-s + (0.258 + 0.965i)37-s + (−0.642 − 0.766i)41-s + (−0.422 + 0.906i)43-s + (0.939 − 0.342i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.819 − 0.573i)5-s + (0.573 + 0.819i)11-s + (0.996 + 0.0871i)13-s + 17-s + (−0.707 − 0.707i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (0.0871 + 0.996i)29-s + (0.939 + 0.342i)31-s + (0.258 + 0.965i)37-s + (−0.642 − 0.766i)41-s + (−0.422 + 0.906i)43-s + (0.939 − 0.342i)47-s + (0.258 + 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.411 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.317299540 + 0.8508051670i\)
\(L(\frac12)\) \(\approx\) \(1.317299540 + 0.8508051670i\)
\(L(1)\) \(\approx\) \(1.031687126 + 0.08883345173i\)
\(L(1)\) \(\approx\) \(1.031687126 + 0.08883345173i\)

Euler product

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

−17.58279112042154946531643620235, −16.685128681912557573427793808065, −16.4258259265520997029478687077, −15.548577906333580703640615469354, −14.9935055949846825778913048294, −14.3538303268082304510074194584, −13.76412062150975967738248462321, −12.99383092788568639532094991888, −12.13071438372592805308027874260, −11.68885309664456967210201333369, −10.94628875608332230092202269655, −10.469855162070766507340283580981, −9.68809898418937487893666657091, −8.622913985638252136415406531748, −8.318623692690153340874291718965, −7.60472497867147500922399511906, −6.69541335581536826983563194561, −6.192358087801376348952609286973, −5.5103595096788538539641893508, −4.34458660175865962117378961789, −3.83995443156527319297112623659, −3.21416072854355078081461125412, −2.431349176532267996844535779031, −1.27694194349854464963536312671, −0.47760107054169557382090220680, 1.07104678745447286797223402973, 1.40723614548758458075396323228, 2.69169699969975177525474008134, 3.51878376440425510418745363739, 4.105192750141445287338178781168, 4.8557087334351655834583660428, 5.46024896258441987824169232477, 6.52927359306034698022682401702, 7.03153216231222496741039756678, 7.83162091594204846655654419342, 8.5296541103142102978858782906, 9.03558833890379300977127279465, 9.79121404366275175626434595647, 10.63492728361658899084035724175, 11.2957026763342597427318595532, 11.99895575626023705695685862649, 12.4104368800726936446077179959, 13.237026705902830304943653442370, 13.76323122229006726092135898736, 14.82653056641728819723227598766, 15.13321637143656166973072589151, 15.90029004054217248148774289200, 16.458632583091148743105817043331, 17.182005361323244687538010724389, 17.65201172806957538967661452878

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