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Properties

Label 1-4033-4033.754-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.438 - 0.898i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.342 + 0.939i)2-s + (0.835 − 0.549i)3-s + (−0.766 − 0.642i)4-s + (−0.116 + 0.993i)5-s + (0.230 + 0.973i)6-s + (0.597 − 0.802i)7-s + (0.866 − 0.5i)8-s + (0.396 − 0.918i)9-s + (−0.893 − 0.448i)10-s + (0.893 − 0.448i)11-s + (−0.993 − 0.116i)12-s + (0.802 + 0.597i)13-s + (0.549 + 0.835i)14-s + (0.448 + 0.893i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)2-s + (0.835 − 0.549i)3-s + (−0.766 − 0.642i)4-s + (−0.116 + 0.993i)5-s + (0.230 + 0.973i)6-s + (0.597 − 0.802i)7-s + (0.866 − 0.5i)8-s + (0.396 − 0.918i)9-s + (−0.893 − 0.448i)10-s + (0.893 − 0.448i)11-s + (−0.993 − 0.116i)12-s + (0.802 + 0.597i)13-s + (0.549 + 0.835i)14-s + (0.448 + 0.893i)15-s + (0.173 + 0.984i)16-s + (−0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.438 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.267114205 - 1.416030941i\)
\(L(\frac12)\) \(\approx\) \(2.267114205 - 1.416030941i\)
\(L(1)\) \(\approx\) \(1.279484213 + 0.1129513967i\)
\(L(1)\) \(\approx\) \(1.279484213 + 0.1129513967i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.342 + 0.939i)T \)
3 \( 1 + (0.835 - 0.549i)T \)
5 \( 1 + (-0.116 + 0.993i)T \)
7 \( 1 + (0.597 - 0.802i)T \)
11 \( 1 + (0.893 - 0.448i)T \)
13 \( 1 + (0.802 + 0.597i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (0.998 - 0.0581i)T \)
31 \( 1 + (-0.116 - 0.993i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.686 - 0.727i)T \)
53 \( 1 + (-0.597 + 0.802i)T \)
59 \( 1 + (0.448 + 0.893i)T \)
61 \( 1 + (0.230 + 0.973i)T \)
67 \( 1 + (0.396 - 0.918i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.893 + 0.448i)T \)
79 \( 1 + (-0.116 - 0.993i)T \)
83 \( 1 + (0.893 + 0.448i)T \)
89 \( 1 + (0.727 - 0.686i)T \)
97 \( 1 + (-0.802 - 0.597i)T \)
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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

−18.58534769609840308592495184520, −17.73955019746304667542421738777, −17.30261896088244051310064311683, −16.231626217855208068161785494079, −15.88734140358174995398777286577, −14.89369108555091128481412283349, −14.328290190590103933768887795114, −13.4794880251785649689791595895, −12.88618761992673842226595506015, −12.23041237280336588837787547469, −11.520622493817887843928264181970, −10.8587723855488961079499183763, −9.98541675260417040699252016147, −9.28927870195360451839330872306, −8.804420410728792127168788613514, −8.31778601561124374734266259813, −7.76439663938537736224077706706, −6.48860517778364894801473190416, −5.13661543830022732902690479908, −4.84017148068830153792125354655, −4.01006776503099069915261324934, −3.309538148979694388639615074, −2.46615553269467750068578220067, −1.53658980843576730454171534304, −1.153755787721272575909298362483, 0.4017903148445482707418762271, 1.21013717719040028858789176958, 2.01698043989987844399001264875, 3.14553014871708363099055462072, 3.99161352800589431240197589048, 4.44994080018683926333084736581, 5.79312666894780410778885047526, 6.61780082895655322744522798122, 6.9985695688420936582797588979, 7.46000845716997505083072307955, 8.54760699380815364916432681876, 8.77270792120578017366014493291, 9.635789301174521780557646791, 10.55227727910977187715344323664, 11.171410532090582041852239900542, 11.88093889667813816018348933440, 13.32584700540388408169380111927, 13.64792005371471564160992535758, 14.02821793753077128699417015333, 14.90002394832578575104109538974, 15.21493389205780238414006666632, 16.02169205449434811967486822039, 17.00134942849585633826492568873, 17.565678548628325998392847779244, 18.10431152293755684276869843860

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