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Properties

Label 1-4033-4033.2635-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.867 + 0.498i$
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 + (−0.173 + 0.984i)3-s + (−0.5 + 0.866i)4-s + (−0.342 − 0.939i)5-s + (0.939 − 0.342i)6-s + (−0.939 + 0.342i)7-s + 8-s + (−0.939 − 0.342i)9-s + (−0.642 + 0.766i)10-s + (0.642 + 0.766i)11-s + (−0.766 − 0.642i)12-s + (−0.939 + 0.342i)13-s + (0.766 + 0.642i)14-s + (0.984 − 0.173i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.173 + 0.984i)3-s + (−0.5 + 0.866i)4-s + (−0.342 − 0.939i)5-s + (0.939 − 0.342i)6-s + (−0.939 + 0.342i)7-s + 8-s + (−0.939 − 0.342i)9-s + (−0.642 + 0.766i)10-s + (0.642 + 0.766i)11-s + (−0.766 − 0.642i)12-s + (−0.939 + 0.342i)13-s + (0.766 + 0.642i)14-s + (0.984 − 0.173i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.867 + 0.498i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2635, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.867 + 0.498i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1172458963 + 0.4394027947i\)
\(L(\frac12)\) \(\approx\) \(0.1172458963 + 0.4394027947i\)
\(L(1)\) \(\approx\) \(0.5718107952 + 0.06129347014i\)
\(L(1)\) \(\approx\) \(0.5718107952 + 0.06129347014i\)

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 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (0.642 + 0.766i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.642 + 0.766i)T \)
31 \( 1 + (-0.984 + 0.173i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.342 - 0.939i)T \)
53 \( 1 + (-0.642 + 0.766i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (-0.984 - 0.173i)T \)
97 \( 1 + (0.984 + 0.173i)T \)
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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

−18.37169732354324617599287478367, −17.388389285880756072755143684930, −16.94859823154346981202211282340, −16.251976365838660916979515635246, −15.60507519529652889115450656382, −14.640130214738969033166647620003, −14.16496122877795605074572556502, −13.68296783423526427810115329740, −12.80357419145403942138224986214, −12.02547332858495083048384447401, −11.20756603921133240780799160536, −10.61362059096232210372704317979, −9.64995286661183097913101067120, −9.21430998938365968333076667111, −8.07659280691815471377740473117, −7.52346727870287917044036909520, −7.0077150233251293649016392137, −6.47046232510708468183868869459, −5.77867530151315782036452684318, −5.03834951141650366196241385873, −3.72861401362702957792046733576, −3.02750870317689372126617882780, −2.13341493440849706831991189422, −0.82299120554976373185216760456, −0.230334991826207036753164946829, 1.093774117074994972212697838093, 1.98173237449581735493983407272, 3.14718815119818598557183442365, 3.628136995288711191487808833371, 4.36593851508400506604999271789, 5.09668092430416281652623995080, 5.7700893090525181387990800574, 7.089402006893940074747063210455, 7.70033001033581213821908110818, 8.92979504098307007406110215728, 9.0792863577808789895526379576, 9.76859645956429017810613949173, 10.21929345952350785324752282318, 11.275335318432036090137559880688, 11.88447478364443604231024731520, 12.39118770624536116119043366206, 12.927864560951895052017312803945, 13.89577180269977612961537905230, 14.84141006987888584410352777472, 15.447823044299002231926759479028, 16.42439861472207230263462873185, 16.639897589108708538796344338, 17.26060169899374971447581565912, 17.99007822944347838287473157242, 19.01837763211338322523851327079

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