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Properties

Label 1-4033-4033.2314-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.989 - 0.143i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.866 − 0.5i)2-s + (0.0581 + 0.998i)3-s + (0.5 − 0.866i)4-s + (0.918 − 0.396i)5-s + (0.549 + 0.835i)6-s + (−0.993 − 0.116i)7-s − i·8-s + (−0.993 + 0.116i)9-s + (0.597 − 0.802i)10-s + (−0.597 − 0.802i)11-s + (0.893 + 0.448i)12-s + (0.918 − 0.396i)13-s + (−0.918 + 0.396i)14-s + (0.448 + 0.893i)15-s + (−0.5 − 0.866i)16-s − i·17-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.0581 + 0.998i)3-s + (0.5 − 0.866i)4-s + (0.918 − 0.396i)5-s + (0.549 + 0.835i)6-s + (−0.993 − 0.116i)7-s − i·8-s + (−0.993 + 0.116i)9-s + (0.597 − 0.802i)10-s + (−0.597 − 0.802i)11-s + (0.893 + 0.448i)12-s + (0.918 − 0.396i)13-s + (−0.918 + 0.396i)14-s + (0.448 + 0.893i)15-s + (−0.5 − 0.866i)16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.989 - 0.143i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.989 - 0.143i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2463265412 - 3.411863260i\)
\(L(\frac12)\) \(\approx\) \(0.2463265412 - 3.411863260i\)
\(L(1)\) \(\approx\) \(1.591392777 - 0.7722976694i\)
\(L(1)\) \(\approx\) \(1.591392777 - 0.7722976694i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.0581 + 0.998i)T \)
5 \( 1 + (0.918 - 0.396i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (-0.597 - 0.802i)T \)
13 \( 1 + (0.918 - 0.396i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (-0.918 + 0.396i)T \)
31 \( 1 + (0.957 - 0.286i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.835 - 0.549i)T \)
53 \( 1 + (-0.0581 - 0.998i)T \)
59 \( 1 + (-0.448 - 0.893i)T \)
61 \( 1 + (0.230 + 0.973i)T \)
67 \( 1 + (-0.893 + 0.448i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.396 + 0.918i)T \)
79 \( 1 + (-0.549 + 0.835i)T \)
83 \( 1 + (-0.686 - 0.727i)T \)
89 \( 1 + (-0.918 + 0.396i)T \)
97 \( 1 + (0.957 + 0.286i)T \)
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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

−18.48171099957604715576331170917, −17.9335535160414934093879880312, −17.18944279342125160005946226253, −16.680794192985910053342301572, −15.776956668752643892764302204547, −15.10355211205011003926667263096, −14.412184623263792856400455560898, −13.64490388268404456437021650901, −13.24621165112914303260133240215, −12.76408770189457801145327385125, −12.095221060977404337489511715399, −11.24256989620839563558597264278, −10.45612099342108919411493320425, −9.52446159665821033842634170142, −8.79756362103892893821735119569, −7.799872937019595886688867787837, −7.30514517414592223121149156606, −6.444588451991264269444152472175, −6.051754856508444502639361271831, −5.57412060142892891360000215973, −4.463107454510082672815245043794, −3.30766751506439066304915992624, −2.965750361857916342208108025507, −1.948431478069321199693368407290, −1.38640166886427902495395198312, 0.351167965870073969422715464356, 0.953228334236400389687750387688, 2.40654892326837733336781021200, 2.939039962086682537972444503502, 3.49643084639231647873790364601, 4.44624847098674356826893353253, 5.240943451287149774644042036537, 5.667244195731717303002744352029, 6.33314932886377125977441871406, 7.2111258033995434043129159032, 8.6564103742799633836933204379, 9.08521090135807493562050608320, 9.93836680140060535918544409339, 10.29801141240914499417896062882, 11.10348032135466053613538006299, 11.6294582286065476460616667388, 12.82905881925854590122674438959, 13.18432296719758245616511102489, 13.83847001033571635512470364073, 14.3155888128206526473222716727, 15.4021724449786060424838563296, 15.90012287931308255298678200371, 16.30667703545624045205715903272, 17.04546533769294478642125586661, 18.094153631354439807948602132720

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