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Properties

Label 1-4033-4033.1799-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.999 + 0.000318i$
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  + (−0.5 + 0.866i)2-s + (−0.766 + 0.642i)3-s + (−0.5 − 0.866i)4-s + (−0.984 + 0.173i)5-s + (−0.173 − 0.984i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s + (0.939 + 0.342i)12-s + (0.173 + 0.984i)13-s + (−0.939 − 0.342i)14-s + (0.642 − 0.766i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.766 + 0.642i)3-s + (−0.5 − 0.866i)4-s + (−0.984 + 0.173i)5-s + (−0.173 − 0.984i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s + (0.939 + 0.342i)12-s + (0.173 + 0.984i)13-s + (−0.939 − 0.342i)14-s + (0.642 − 0.766i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.999 + 0.000318i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.999 + 0.000318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5402509509 + 8.593305000\times10^{-5}i\)
\(L(\frac12)\) \(\approx\) \(0.5402509509 + 8.593305000\times10^{-5}i\)
\(L(1)\) \(\approx\) \(0.4781001704 + 0.2624307241i\)
\(L(1)\) \(\approx\) \(0.4781001704 + 0.2624307241i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.342 - 0.939i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.342 - 0.939i)T \)
31 \( 1 + (-0.642 + 0.766i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.984 + 0.173i)T \)
53 \( 1 + (0.342 - 0.939i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.342 - 0.939i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 + (0.642 + 0.766i)T \)
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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

−18.419944538167620787889272397558, −17.87106465288707951447164014693, −17.257444328270786396716705909909, −16.719525984406206637757781319, −15.991065866849099855775473078048, −15.20327993560589797914969792868, −14.17297289081348400510871917426, −13.25581165740686924455186497019, −12.833347773814530777688349807093, −12.22506800079728252750062541930, −11.60249565079105731999082192944, −10.84633774311663324162666374171, −10.48025019879362312818994199708, −9.73136045112435461517632067342, −8.61870999478072185810514999212, −7.7957385163957834095858124910, −7.4799289087569362286856860348, −6.976840060909138835394707478151, −5.5129112076305575792714893075, −4.94927969947213725587932495720, −4.011523335500524054905408874299, −3.46964854033160052549321225123, −2.407522756103227697221493441036, −1.255554446438222066073543363271, −0.920664458485966946245043739546, 0.305839941358278785979927152041, 1.19325355277571643500403069321, 2.69444363251435373913478530416, 3.58678329994771412372360646096, 4.53425353933786065794987332207, 5.107381404986118849969049231482, 5.76213692141563399107171080470, 6.52709173399925478468451579475, 7.18118955144965838511542239021, 8.08219550567882735290105140439, 8.73726548136317241476374506230, 9.32193420544954525876774827492, 10.067336849730510936788441470332, 11.01476515189551668980218022784, 11.44745638488160252849188840627, 12.00605994224230946494193818570, 12.96094306653387633546141669308, 14.166069979694772430525702730822, 14.49861796985215003490956128687, 15.476728977140161057024491711898, 15.79004409900410236781557760097, 16.433071299304785913150897875066, 16.71592466470250764524813812546, 17.919922258552851294507535919665, 18.3942061933353177750213651952

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