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Properties

Label 1-4033-4033.1136-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.426 - 0.904i$
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 + 3-s + (−0.5 − 0.866i)4-s + 5-s + (0.5 − 0.866i)6-s + 7-s − 8-s + 9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s + 15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + 5-s + (0.5 − 0.866i)6-s + 7-s − 8-s + 9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s − 13-s + (0.5 − 0.866i)14-s + 15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.426 - 0.904i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.426 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.978505640 - 2.523087806i\)
\(L(\frac12)\) \(\approx\) \(3.978505640 - 2.523087806i\)
\(L(1)\) \(\approx\) \(2.171477018 - 1.044597515i\)
\(L(1)\) \(\approx\) \(2.171477018 - 1.044597515i\)

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

−18.43940112466435402349536696460, −17.704630218653434035834796876933, −17.26320726078579082063601265774, −16.54224955592791194897155115044, −15.71858675911298798571699114176, −14.95559482091495554767296426608, −14.34604486639578235456654170218, −14.085558701943948677581578980777, −13.46831819806275126459096241531, −12.6798081214368210901074032030, −12.01622341027736411619739618153, −11.02468465278794676192149134341, −10.02669629218996162534825855668, −9.39229037118109699366211802520, −8.65369364759992799382173640825, −8.19997624029362890532533232136, −7.340869155411883726192778424066, −6.80441152169507009849083447759, −5.686746505454208865430887762758, −5.35470336823315219648712677666, −4.30140758979248375663929761764, −3.7527606834869448161521097047, −2.68748392632237719481656860169, −2.12734768912517944795546826823, −1.0603075599961519985198515673, 1.168223288909045403472554132619, 1.783553392109835524113148001188, 2.31970565758511965548932142296, 3.05302136749025468728538486028, 4.079603396853694830465273712196, 4.67134348497662468508796978297, 5.362640784644495124703307051538, 6.18396501330825177928400559926, 7.3696821992118686424530204745, 7.78477483540908443900171357452, 9.07878810856647262141693050140, 9.36433040135270447985847377990, 10.02235930568977662972229797932, 10.58993543666138472767883648620, 11.64570980814097859358310736949, 12.31779257178273661364931788725, 12.80535430884149887876204777243, 13.80899588494237138650257897699, 14.19950503596758882599780296552, 14.606929391005138631676927706381, 15.134658274127036219757304346350, 16.26539991156927223703170837167, 17.19717362641971531484211534359, 18.04009450209452881389242554393, 18.30662794283079234299011403791

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