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Properties

Label 1-4033-4033.3370-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: $\chi_{4033} (3370, \cdot )$
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

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

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