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Properties

Label 1-4033-4033.3781-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.284 - 0.958i$
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  + (−0.939 − 0.342i)2-s + (−0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + 5-s + (0.766 − 0.642i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (−0.939 + 0.342i)12-s + (−0.5 + 0.866i)13-s + (−0.939 − 0.342i)14-s + (−0.5 + 0.866i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + 5-s + (0.766 − 0.642i)6-s + 7-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (−0.939 − 0.342i)10-s + (−0.939 + 0.342i)11-s + (−0.939 + 0.342i)12-s + (−0.5 + 0.866i)13-s + (−0.939 − 0.342i)14-s + (−0.5 + 0.866i)15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.284 - 0.958i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.284 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6502129411 - 0.4853383274i\)
\(L(\frac12)\) \(\approx\) \(0.6502129411 - 0.4853383274i\)
\(L(1)\) \(\approx\) \(0.7059622677 + 0.003753982557i\)
\(L(1)\) \(\approx\) \(0.7059622677 + 0.003753982557i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−18.34259121594903297643741150765, −17.850502058147777881933618607676, −17.5802034235922697938724713319, −16.6378026400950717190098729051, −16.37603885114716231114331513703, −15.160715696565899725755479911574, −14.518996886033754084426689095420, −13.98002399745287889745302976327, −13.0620142072541281020448490236, −12.448899697554694150469194803809, −11.573670467596245126175272060308, −10.92495491458784935518019604716, −10.223387618162418330222507487171, −9.84266965382461374995746347893, −8.5792749080990299489555515873, −8.021961985087645551322422039208, −7.657789807949752733369053217552, −6.76303117101493148000569270239, −5.77674812136875348063512956715, −5.57832851096031234522450088126, −4.9383203484367275688073483314, −3.15035389091628776081756457046, −2.33250130865976627673277420022, −1.56338658204932150957825804327, −1.105288897448887321919329616933, 0.33924827630692438556098303588, 1.47276103969237695933459874241, 2.342157471300789656365524807569, 2.88795642477975068659714465073, 4.14137831664541001826199185658, 4.9321943491015150528537256158, 5.45254496333510349952472739388, 6.45840627439853274553846707649, 7.1696395579450000299963894758, 8.00554841812801772678350793982, 8.951054477439850486803724160684, 9.321384425140830045834350518629, 10.204280725333893336199940208336, 10.492463439390791188809208962715, 11.25244699996741677550412823769, 11.97096352822211245634859949952, 12.49952599721963515563102967688, 13.6588166812369013569451255901, 14.32538579250198648526956315073, 15.096990863704381906574840216134, 15.80739889337727424341095980422, 16.56085525818239115060682510003, 17.02099749360087189459840388064, 17.717128481084822600998706415352, 18.15106413083611064478140025584

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