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Properties

Label 1-4033-4033.166-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.644 - 0.764i$
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 + (−0.893 + 0.448i)3-s + 4-s + (0.116 + 0.993i)5-s + (0.893 − 0.448i)6-s + (0.597 + 0.802i)7-s − 8-s + (0.597 − 0.802i)9-s + (−0.116 − 0.993i)10-s + (0.116 − 0.993i)11-s + (−0.893 + 0.448i)12-s + (0.993 − 0.116i)13-s + (−0.597 − 0.802i)14-s + (−0.549 − 0.835i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (−0.893 + 0.448i)3-s + 4-s + (0.116 + 0.993i)5-s + (0.893 − 0.448i)6-s + (0.597 + 0.802i)7-s − 8-s + (0.597 − 0.802i)9-s + (−0.116 − 0.993i)10-s + (0.116 − 0.993i)11-s + (−0.893 + 0.448i)12-s + (0.993 − 0.116i)13-s + (−0.597 − 0.802i)14-s + (−0.549 − 0.835i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.644 - 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6421496024 - 0.2984834153i\)
\(L(\frac12)\) \(\approx\) \(0.6421496024 - 0.2984834153i\)
\(L(1)\) \(\approx\) \(0.5963742196 + 0.07458450244i\)
\(L(1)\) \(\approx\) \(0.5963742196 + 0.07458450244i\)

Euler product

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

−18.25432326787727313282512593707, −17.655932735854008016985241290869, −17.46139576027937765719824485327, −16.60495482500573407939918072296, −16.27821806207141997734970512199, −15.4681197747003104626762617650, −14.60023265080416228924101231485, −13.59024306067382570335338410754, −12.8645090058709199896182611111, −12.28192328218346561591280573258, −11.660473114859059520748419569945, −10.7253594069751607784926458138, −10.54232801488128775679508353122, −9.568899728061817822161991029287, −8.77184957610690653591382407532, −8.054865759284532878083669954030, −7.51279543999078926852670503903, −6.67765129848124956034232728366, −6.090437383408332250247676913914, −5.18433082927408803442088770770, −4.47025441520085534943653450026, −3.61026434926005581806043138623, −2.03956128477561549686927494545, −1.389310417913503915573232002929, −1.10399188739698432099552620124, 0.367773056915521501273589012557, 1.289548207626890802930635359231, 2.51079659893409777210444825133, 2.96008043943420613786221545043, 4.065438860887847443510287856777, 5.09768986133497699205303785637, 6.11347869407206173244876347710, 6.19421782877061575238521024706, 7.12732566105529551109278414320, 7.93875636933440300401159091432, 8.956729828511574224097221167672, 9.18670190633501389180965583407, 10.25413879436217331354769789115, 10.94329652983596207720387912998, 11.28813404698673133437428073533, 11.634562995036082899780123536, 12.676982908459856855121320835242, 13.66083306561208184104912753284, 14.61178155643646946442262949544, 15.21337372977473373950408559799, 15.932048816546128483987443393258, 16.15680809284228101289986996616, 17.37887266356695960019266640756, 17.65221435900321096648079086500, 18.26974179849501253349771396359

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