Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.623 - 0.781i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.623 - 0.781i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (2765, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ -0.623 - 0.781i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3699113552 - 0.7681907548i$
$L(\frac12,\chi)$  $\approx$  $0.3699113552 - 0.7681907548i$
$L(\chi,1)$  $\approx$  0.6562860785 - 0.07880881983i
$L(1,\chi)$  $\approx$  0.6562860785 - 0.07880881983i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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