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Properties

Label 1-4033-4033.2195-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.181 - 0.983i$
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 + (0.597 − 0.802i)3-s + (−0.5 − 0.866i)4-s + (0.973 − 0.230i)5-s + (0.396 + 0.918i)6-s + (−0.286 + 0.957i)7-s + 8-s + (−0.286 − 0.957i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.993 − 0.116i)12-s + (0.973 − 0.230i)13-s + (−0.686 − 0.727i)14-s + (0.396 − 0.918i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.597 − 0.802i)3-s + (−0.5 − 0.866i)4-s + (0.973 − 0.230i)5-s + (0.396 + 0.918i)6-s + (−0.286 + 0.957i)7-s + 8-s + (−0.286 − 0.957i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.993 − 0.116i)12-s + (0.973 − 0.230i)13-s + (−0.686 − 0.727i)14-s + (0.396 − 0.918i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.181 - 0.983i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.181 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9478687602 - 1.138479937i\)
\(L(\frac12)\) \(\approx\) \(0.9478687602 - 1.138479937i\)
\(L(1)\) \(\approx\) \(1.052009741 - 0.1478155976i\)
\(L(1)\) \(\approx\) \(1.052009741 - 0.1478155976i\)

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 \( 1 + (0.597 - 0.802i)T \)
5 \( 1 + (0.973 - 0.230i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (-0.286 - 0.957i)T \)
13 \( 1 + (0.973 - 0.230i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.686 - 0.727i)T \)
31 \( 1 + (-0.835 - 0.549i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (0.396 - 0.918i)T \)
53 \( 1 + (0.396 + 0.918i)T \)
59 \( 1 + (0.396 - 0.918i)T \)
61 \( 1 + (-0.835 + 0.549i)T \)
67 \( 1 + (-0.993 + 0.116i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.686 + 0.727i)T \)
79 \( 1 + (0.597 + 0.802i)T \)
83 \( 1 + (-0.835 + 0.549i)T \)
89 \( 1 + (-0.286 + 0.957i)T \)
97 \( 1 + (-0.835 + 0.549i)T \)
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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

−18.69747278947085830405432591002, −17.80546498327704214047340906155, −17.60209945551850095181979400145, −16.607857476296497312463153628019, −16.20972288974489585670207805050, −15.22190745381621200485225507954, −14.43906124962673042559860657107, −13.67950284822088585973769507083, −13.23647771436075778367470243944, −12.76196271390582913783055107436, −11.3713667862367523527145235653, −10.88888351743948016342297779035, −10.31843501696338318477260318779, −9.7142137105824150477007293561, −9.20264374910099834040243806028, −8.57860369515916012372637843239, −7.50846579826180640688193878395, −7.07057724919715249032265044828, −5.85828517432916172785904955623, −4.87843195653713530939556833132, −4.238278599525522094687234421531, −3.40671439893515941280320682231, −2.92261618142228099091105112470, −1.78539679280960405653489526392, −1.42401761700886597446141637188, 0.43293260873570224814227882887, 1.34080311877664290083418300937, 2.19662072975382046135319245913, 2.90067466168840206304110697769, 3.96842235981595987254766447579, 5.35987027953420498075733334379, 5.72407630107159611360430451285, 6.294399204512965983293167002104, 7.01773190424700833194938107341, 7.94289401472389676939213711282, 8.57291875503809351690598653232, 9.07353450931845857141110028776, 9.544051855340827426606723277314, 10.50744521988455805292346955702, 11.363974775070920136096079102946, 12.316973969252310178163236368346, 13.21319682869529250955180970061, 13.53597666395292293717085276581, 14.12254525240913333367291105399, 14.90105252518502856467531661001, 15.57666338240877198113632829696, 16.34342603791804047565335510972, 16.830193995519665647860583097553, 17.91832019394608543722461401481, 18.298670269125946034391018235626

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