Properties

Label 1-4033-4033.310-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.987 + 0.158i$
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  − 2-s + (−0.766 − 0.642i)3-s + 4-s + (−0.984 − 0.173i)5-s + (0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s − 8-s + (0.173 + 0.984i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s + (−0.766 − 0.642i)12-s + (−0.173 + 0.984i)13-s + (−0.173 + 0.984i)14-s + (0.642 + 0.766i)15-s + 16-s + 17-s + ⋯
L(s)  = 1  − 2-s + (−0.766 − 0.642i)3-s + 4-s + (−0.984 − 0.173i)5-s + (0.766 + 0.642i)6-s + (0.173 − 0.984i)7-s − 8-s + (0.173 + 0.984i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s + (−0.766 − 0.642i)12-s + (−0.173 + 0.984i)13-s + (−0.173 + 0.984i)14-s + (0.642 + 0.766i)15-s + 16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.987 + 0.158i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (310, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.987 + 0.158i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003210116818 - 0.04034349215i\)
\(L(\frac12)\) \(\approx\) \(0.003210116818 - 0.04034349215i\)
\(L(1)\) \(\approx\) \(0.3904817734 - 0.07684246118i\)
\(L(1)\) \(\approx\) \(0.3904817734 - 0.07684246118i\)

Euler product

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

−18.64689277341835038534948470096, −18.172869093349312003702160921875, −17.5681330086270948848188359581, −16.76941615591278822225621545402, −15.99336120336794169304418705102, −15.639271971317191375865710312289, −15.167085059694628926682113069771, −14.55275913591862250754915287434, −12.96840459367623016603683260985, −12.39127196180566864498910847229, −11.63913711467766250853924290563, −11.25582309168218642449814346282, −10.604004211981952467403374879075, −9.817607994235277505007218480116, −9.27528371031200901772403935040, −8.327424890578320591525357317, −7.75329327437730054278536027810, −7.18802004462792101114632895458, −6.02147275401176534375304549476, −5.51108048518220582738892503000, −4.88922838054180829720082502582, −3.4925867412659561982626274014, −3.13587711349178851492478225218, −2.09223161007033999279316763024, −0.770659834936088500367549824832, 0.02806080637984758846425485781, 1.05319074072686035784058347404, 1.67130093138012114693040569135, 2.761047141479885049783746143814, 3.77104400680113924351312788374, 4.6363825563673502154119284023, 5.51632892353184069912868631856, 6.37454923804867664299689543152, 7.23860098239646253562422151693, 7.6170332423232559682840120493, 8.010989973782721010733315839612, 8.98684603396511427889242522801, 10.07096533123287044700647702120, 10.52436016137180195465893607725, 11.160288485046424032512319387183, 11.89690790802807227390346367779, 12.31838618894521838890609285283, 13.07186151105530499241723734325, 14.134256464457959801133520209048, 14.74069436428132519768325260311, 15.86547696051638230020620611958, 16.40518061542299287168959470570, 16.629031396295112739024785764802, 17.42768869687111089615408887336, 18.25415495544856913893764881635

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