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Properties

Label 1-4033-4033.770-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.184 - 0.982i$
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  − 2-s + (−0.686 + 0.727i)3-s + 4-s + (0.835 + 0.549i)5-s + (0.686 − 0.727i)6-s + (−0.0581 + 0.998i)7-s − 8-s + (−0.0581 − 0.998i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.686 + 0.727i)12-s + (0.835 + 0.549i)13-s + (0.0581 − 0.998i)14-s + (−0.973 + 0.230i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.686 + 0.727i)3-s + 4-s + (0.835 + 0.549i)5-s + (0.686 − 0.727i)6-s + (−0.0581 + 0.998i)7-s − 8-s + (−0.0581 − 0.998i)9-s + (−0.835 − 0.549i)10-s + (−0.835 + 0.549i)11-s + (−0.686 + 0.727i)12-s + (0.835 + 0.549i)13-s + (0.0581 − 0.998i)14-s + (−0.973 + 0.230i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.184 - 0.982i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (770, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.184 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1344188824 + 0.1619517792i\)
\(L(\frac12)\) \(\approx\) \(-0.1344188824 + 0.1619517792i\)
\(L(1)\) \(\approx\) \(0.4696321188 + 0.2937801196i\)
\(L(1)\) \(\approx\) \(0.4696321188 + 0.2937801196i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.686 + 0.727i)T \)
5 \( 1 + (0.835 + 0.549i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (-0.835 + 0.549i)T \)
13 \( 1 + (0.835 + 0.549i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.0581 - 0.998i)T \)
31 \( 1 + (-0.396 - 0.918i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.686 - 0.727i)T \)
53 \( 1 + (-0.286 - 0.957i)T \)
59 \( 1 + (-0.973 + 0.230i)T \)
61 \( 1 + (0.993 - 0.116i)T \)
67 \( 1 + (-0.686 - 0.727i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.0581 - 0.998i)T \)
79 \( 1 + (-0.973 + 0.230i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (-0.893 + 0.448i)T \)
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

−17.958437264013364741802663899113, −17.36635415319515180872496002507, −16.769476350873550227046322855541, −16.27400780143996122793445331628, −15.72408064701999685849277621176, −14.457784150812839788245325952643, −13.71114243746872001949159505608, −12.97100711855018493433851386396, −12.698911480445630341087649296778, −11.57385230733434272956169799300, −10.77252822410839712673891965672, −10.60941018784110149144739712355, −9.775607055864704766866490446847, −8.759822062829928973475232635604, −8.30831101469119348086805881610, −7.4347576882338980669450708566, −6.810476968561322171301523933691, −6.20071206971469520174725167709, −5.34985998273908786937403905673, −4.81942530795611923384621245157, −3.24264475823479058995670555367, −2.60083618741914341376267069085, −1.39531681512107035557345629176, −1.07686705715344230855993627277, −0.095722391764933535354515807755, 1.528397452294939474010794608410, 2.07853387988580230952626650136, 3.03679426000657120224807730466, 3.809709640270172850187874806130, 5.12711372254077020260719894779, 5.712373099659211368768785205, 6.31929230676631810569542706137, 6.84013291365617424751432155970, 8.048729106190222315648762123161, 8.65240095626740885672608438909, 9.54931212444337547543825172850, 9.86767243428270282281376234225, 10.61776650254598145415599049397, 11.13518903744539675640060204548, 11.851382311315918473445052841117, 12.61317993205723958516200881524, 13.359360614569584777229412333, 14.670403135251813411114898011062, 15.160642239401268537626241838006, 15.47332832016304361901861258, 16.568584324014137626426589571567, 16.86283247009426178623224215342, 17.66610688398253403219076256533, 18.24454731550389360724051689552, 18.72261990441716512093402737978

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