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Properties

Label 1-4033-4033.2242-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.569 + 0.821i$
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 + (−0.893 − 0.448i)3-s + (−0.5 − 0.866i)4-s + (0.116 − 0.993i)5-s + (0.835 − 0.549i)6-s + (0.597 − 0.802i)7-s + 8-s + (0.597 + 0.802i)9-s + (0.802 + 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.0581 + 0.998i)12-s + (−0.993 − 0.116i)13-s + (0.396 + 0.918i)14-s + (−0.549 + 0.835i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.893 − 0.448i)3-s + (−0.5 − 0.866i)4-s + (0.116 − 0.993i)5-s + (0.835 − 0.549i)6-s + (0.597 − 0.802i)7-s + 8-s + (0.597 + 0.802i)9-s + (0.802 + 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.0581 + 0.998i)12-s + (−0.993 − 0.116i)13-s + (0.396 + 0.918i)14-s + (−0.549 + 0.835i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.569 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3648718529 + 0.1909793808i\)
\(L(\frac12)\) \(\approx\) \(0.3648718529 + 0.1909793808i\)
\(L(1)\) \(\approx\) \(0.5194097889 + 0.007227341954i\)
\(L(1)\) \(\approx\) \(0.5194097889 + 0.007227341954i\)

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 \( 1 + (-0.893 - 0.448i)T \)
5 \( 1 + (0.116 - 0.993i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (-0.802 + 0.597i)T \)
13 \( 1 + (-0.993 - 0.116i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.918 + 0.396i)T \)
31 \( 1 + (0.957 - 0.286i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (0.549 - 0.835i)T \)
53 \( 1 + (-0.549 - 0.835i)T \)
59 \( 1 + (-0.835 - 0.549i)T \)
61 \( 1 + (-0.957 - 0.286i)T \)
67 \( 1 + (-0.998 - 0.0581i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.396 + 0.918i)T \)
79 \( 1 + (0.893 - 0.448i)T \)
83 \( 1 + (0.286 - 0.957i)T \)
89 \( 1 + (-0.802 - 0.597i)T \)
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

−18.20344820217715324680080177515, −17.93268873641747162141087862101, −17.20131862523450240132673085851, −16.568840659970892338708859834264, −15.56898050154846911657695226386, −15.21868090402941295129192008586, −14.147928439069278967995766065756, −13.598447700860302840094858408132, −12.35960391390640357030190985543, −12.13824343898697996807350911812, −11.26424358485446487586895448556, −10.8968950858126636023191565766, −10.23447684014210579785257297470, −9.54724447544761538685237835817, −8.94950688514263477050396443628, −7.75215505262701280742510840187, −7.45140029260981040528963491532, −6.2696732812216256459811721070, −5.56255866921269115367384113773, −4.81326471963073866170268058010, −4.06006977930075814511990160709, −2.98565631512921142725393363691, −2.5225130128328593157684376155, −1.608771632677088923099218605114, −0.25288238444924502522302443988, 0.60712647938325583781366571504, 1.59943006377576012149465259059, 2.10336092424419424204696849402, 4.126119968381935892332711285122, 4.61923836000071835689965687211, 5.19876568868355547241760511419, 5.91053682150676142859857824129, 6.63578365987554927169702763126, 7.63142572676399290407353374773, 7.83290040411293789535797826665, 8.55306847303956453161735633844, 9.71883863418564222412489461752, 10.253928555483850195742357697039, 10.71524759318338357690919084333, 11.77246034936774229201711792238, 12.63089664672317835063283998214, 12.973554650270838453812312801187, 13.82252745231106198567694429953, 14.6027477304689545958766359783, 15.3063859128259349256874250696, 16.12990570861607774725022964591, 16.864769667732214951233328443946, 17.0705298251858989750583529398, 17.622817370852962947419997954697, 18.32573464547741338216310494190

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