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Properties

Label 1-4033-4033.1320-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.968 + 0.248i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (0.766 − 0.642i)2-s + (0.597 + 0.802i)3-s + (0.173 − 0.984i)4-s + (0.286 + 0.957i)5-s + (0.973 + 0.230i)6-s + (−0.686 + 0.727i)7-s + (−0.5 − 0.866i)8-s + (−0.286 + 0.957i)9-s + (0.835 + 0.549i)10-s + (0.835 − 0.549i)11-s + (0.893 − 0.448i)12-s + (0.973 + 0.230i)13-s + (−0.0581 + 0.998i)14-s + (−0.597 + 0.802i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.597 + 0.802i)3-s + (0.173 − 0.984i)4-s + (0.286 + 0.957i)5-s + (0.973 + 0.230i)6-s + (−0.686 + 0.727i)7-s + (−0.5 − 0.866i)8-s + (−0.286 + 0.957i)9-s + (0.835 + 0.549i)10-s + (0.835 − 0.549i)11-s + (0.893 − 0.448i)12-s + (0.973 + 0.230i)13-s + (−0.0581 + 0.998i)14-s + (−0.597 + 0.802i)15-s + (−0.939 − 0.342i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1320, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.057881989 + 0.5130476524i\)
\(L(\frac12)\) \(\approx\) \(4.057881989 + 0.5130476524i\)
\(L(1)\) \(\approx\) \(2.169691617 + 0.06299317982i\)
\(L(1)\) \(\approx\) \(2.169691617 + 0.06299317982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (0.597 + 0.802i)T \)
5 \( 1 + (0.286 + 0.957i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.835 - 0.549i)T \)
13 \( 1 + (0.973 + 0.230i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.835 + 0.549i)T \)
31 \( 1 + (0.993 + 0.116i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (0.286 - 0.957i)T \)
53 \( 1 + (-0.893 + 0.448i)T \)
59 \( 1 + (0.597 - 0.802i)T \)
61 \( 1 + (-0.893 + 0.448i)T \)
67 \( 1 + (0.835 - 0.549i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (0.893 + 0.448i)T \)
79 \( 1 + (-0.0581 - 0.998i)T \)
83 \( 1 + (0.973 + 0.230i)T \)
89 \( 1 + (-0.597 - 0.802i)T \)
97 \( 1 + (-0.396 + 0.918i)T \)
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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

−18.25060841623613410524981113907, −17.52322483605777978561329908942, −17.07592001926505168501335051520, −16.39350120280172827424328489325, −15.66741964215185671964170665136, −14.99401945460890223607549486148, −14.05944655200059954316021759211, −13.67756980804968717949725862629, −13.151709693172171577762359986655, −12.49629344273861258620770794779, −12.021746066068450357718760987666, −11.16860395772828766191974632934, −9.72494050265310550907949169929, −9.4274580871499666721067527251, −8.32626852786861259239978031048, −8.017589739273051368517197009358, −7.13657361261069327017855109769, −6.404516480032658005474246137632, −5.9915815115921888920670717033, −4.973319205826480406127786106941, −4.105476251100715496094657298923, −3.47715822778230551380433624789, −2.80593498281708906677015779118, −1.46681944976249639631919397478, −1.02445619509289819034527081022, 1.02451941046816983460922966181, 2.12913120794875507788852743835, 2.94248998814471208306214477268, 3.32152925941444151740022351392, 3.86015876140097656932984889210, 5.03384019511494687697693483100, 5.59063493099192479607456616789, 6.45341674770459937616219078709, 6.93587647896730239210486308436, 8.28721761649509331227285825487, 9.12028977481129457773310488354, 9.562694800098905369359321977036, 10.311792090431668744683223144669, 10.87276691706833330147085865659, 11.656077304154724101216651318980, 12.155389800622220601053888395667, 13.302489153334007548244874459433, 13.91106579370972847812741781791, 14.20027495402505934472682027542, 14.97848283690198759318842921576, 15.67764351390817698014587016363, 16.061847627389855090777074528679, 16.95513824331150257912177106979, 18.21499911469074672123267807967, 18.84929679051959240637663897170

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