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Properties

Label 1-4033-4033.2715-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0191 + 0.999i$
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  + (−0.766 − 0.642i)2-s + (0.686 − 0.727i)3-s + (0.173 + 0.984i)4-s + (−0.448 − 0.893i)5-s + (−0.993 + 0.116i)6-s + (−0.835 − 0.549i)7-s + (0.5 − 0.866i)8-s + (−0.0581 − 0.998i)9-s + (−0.230 + 0.973i)10-s + (0.230 + 0.973i)11-s + (0.835 + 0.549i)12-s + (0.835 + 0.549i)13-s + (0.286 + 0.957i)14-s + (−0.957 − 0.286i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.686 − 0.727i)3-s + (0.173 + 0.984i)4-s + (−0.448 − 0.893i)5-s + (−0.993 + 0.116i)6-s + (−0.835 − 0.549i)7-s + (0.5 − 0.866i)8-s + (−0.0581 − 0.998i)9-s + (−0.230 + 0.973i)10-s + (0.230 + 0.973i)11-s + (0.835 + 0.549i)12-s + (0.835 + 0.549i)13-s + (0.286 + 0.957i)14-s + (−0.957 − 0.286i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0191 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0191 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0191 + 0.999i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2715, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0191 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1820970715 - 0.1856162556i\)
\(L(\frac12)\) \(\approx\) \(-0.1820970715 - 0.1856162556i\)
\(L(1)\) \(\approx\) \(0.5326346821 - 0.4447583255i\)
\(L(1)\) \(\approx\) \(0.5326346821 - 0.4447583255i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (0.686 - 0.727i)T \)
5 \( 1 + (-0.448 - 0.893i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (0.230 + 0.973i)T \)
13 \( 1 + (0.835 + 0.549i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.727 + 0.686i)T \)
31 \( 1 + (-0.998 + 0.0581i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.918 + 0.396i)T \)
53 \( 1 + (-0.998 - 0.0581i)T \)
59 \( 1 + (0.286 - 0.957i)T \)
61 \( 1 + (0.802 - 0.597i)T \)
67 \( 1 + (-0.448 + 0.893i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.973 + 0.230i)T \)
79 \( 1 + (0.835 - 0.549i)T \)
83 \( 1 + (0.686 - 0.727i)T \)
89 \( 1 + (-0.802 + 0.597i)T \)
97 \( 1 + (-0.448 - 0.893i)T \)
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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

−19.0207875345278080340895622364, −18.28573643818853977549562976882, −17.84250115125096109434792734123, −16.56457104464697847789581226464, −16.15242767687561592091679390165, −15.67073352554528235382365952, −15.079805687392703409766828972313, −14.53911516114054120892875855748, −13.585842912408150040884298300139, −13.317136268844497923350365529202, −11.72739768522577608651878221867, −11.15660561359867655646623029435, −10.61699031694991039801415025525, −9.75482940561078816909185949811, −9.30136762966784080053137408477, −8.52453806821845353298836286636, −8.00561903009565179477062259809, −7.18839295722479279457047763550, −6.36911675427750454799079483695, −5.81121009860534389276052288883, −4.92844389682436216889311890889, −3.79033411569552879945321886435, −3.11300406619868853268353816084, −2.600536459484985408533392022029, −1.325674471951962213756434937424, 0.09225932157814953171089268876, 1.142108913791585969595836103712, 1.666726635118497259694914588064, 2.6369044430611982898058948330, 3.56784673211659828747952264246, 3.99734994787018815581682790212, 4.92604936715805618755693215757, 6.44253247197797829969190616001, 6.96921116859067015500471846901, 7.49809932126238378570410641724, 8.42829206890262673833524232284, 8.99608091167787268270994933007, 9.351245996180108567743985332071, 10.23157877869847808864617998187, 11.152482140112136365097258540366, 11.847683573023895424214984552140, 12.618015871714654283224496712914, 12.94788289424920638004899824084, 13.52694419482871461122779465516, 14.405842117861789264473663492386, 15.54215965529303548158477503523, 15.9909145869167637318311448973, 16.70666833217174454231201423111, 17.54101509244718947013342650770, 17.91987827273424572431950924763

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