Properties

Label 1-4033-4033.2765-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.623 - 0.781i$
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  + (−0.984 − 0.173i)2-s + (−0.0581 + 0.998i)3-s + (0.939 + 0.342i)4-s + (−0.597 − 0.802i)5-s + (0.230 − 0.973i)6-s + (0.396 − 0.918i)7-s + (−0.866 − 0.5i)8-s + (−0.993 − 0.116i)9-s + (0.448 + 0.893i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.918 + 0.396i)13-s + (−0.549 + 0.835i)14-s + (0.835 − 0.549i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (−0.0581 + 0.998i)3-s + (0.939 + 0.342i)4-s + (−0.597 − 0.802i)5-s + (0.230 − 0.973i)6-s + (0.396 − 0.918i)7-s + (−0.866 − 0.5i)8-s + (−0.993 − 0.116i)9-s + (0.448 + 0.893i)10-s + (0.448 − 0.893i)11-s + (−0.396 + 0.918i)12-s + (0.918 + 0.396i)13-s + (−0.549 + 0.835i)14-s + (0.835 − 0.549i)15-s + (0.766 + 0.642i)16-s + (0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.623 - 0.781i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.623 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3699113552 - 0.7681907548i\)
\(L(\frac12)\) \(\approx\) \(0.3699113552 - 0.7681907548i\)
\(L(1)\) \(\approx\) \(0.6562860785 - 0.07880881983i\)
\(L(1)\) \(\approx\) \(0.6562860785 - 0.07880881983i\)

Euler product

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

−18.60095194350175993612316047718, −17.88003661033285508109839484972, −17.47766295756853941683698785820, −16.79401323659780862846016572997, −15.66453817675059081476190175606, −15.174436124504586295128594271719, −14.75714012667811680732833405008, −13.930785102933442430250595572614, −12.78428062885683719313039872527, −12.23649963133437140051924169914, −11.608901207501770287002258899189, −10.93310351973156463095308872125, −10.48845235116413204805814426723, −9.20886743444698557545414642203, −8.75842360700633133408954409109, −8.009892688060030528488019635189, −7.37891919923789360559516132735, −6.88030970043368168545202952388, −5.985153898491631011709039444662, −5.62250889349628234240929692933, −4.194820594767122843214412026482, −3.06722356303258947306895371408, −2.433564339847226711277417539296, −1.69020067283907004879160833897, −0.87791468130987308921405591715, 0.23949910042515542742138429398, 0.9397044798594293281291777238, 1.67138329669606952236476475407, 3.22233311388377462748225490966, 3.570918631232295707061970533, 4.35379508661250275552272247792, 5.23002664142960327164212344873, 6.09881935708974535248381103907, 6.98464930616030284179318454983, 7.9232773224592799227152771717, 8.404162728059059732338335083410, 9.096800296120928508647820341222, 9.557028936708225985115443214690, 10.62761195011271046121933449813, 11.0268622925514957445970138714, 11.49116635763332894241129558539, 12.31151626736654473540556600212, 13.200938845798396794193025983513, 14.19943019593867133321917532389, 14.75595868051151241767677849104, 15.68388147489828560654731388283, 16.39382127522753252098912131365, 16.61361226828068390623982453739, 17.01905640128718024603046998331, 18.02555063974860422876838117297

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