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Properties

Label 1-4033-4033.2907-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.872 - 0.488i$
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 + (0.993 + 0.116i)6-s + (−0.286 + 0.957i)7-s − 8-s + (−0.286 − 0.957i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (0.396 + 0.918i)12-s + (−0.973 + 0.230i)13-s + (−0.973 + 0.230i)14-s + (−0.396 + 0.918i)15-s + (−0.5 − 0.866i)16-s − 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 + (0.993 + 0.116i)6-s + (−0.286 + 0.957i)7-s − 8-s + (−0.286 − 0.957i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (0.396 + 0.918i)12-s + (−0.973 + 0.230i)13-s + (−0.973 + 0.230i)14-s + (−0.396 + 0.918i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.872 - 0.488i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.872 - 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3124812424 - 0.08161110311i\)
\(L(\frac12)\) \(\approx\) \(0.3124812424 - 0.08161110311i\)
\(L(1)\) \(\approx\) \(0.7584272358 + 0.4107642655i\)
\(L(1)\) \(\approx\) \(0.7584272358 + 0.4107642655i\)

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 \( 1 + (-0.686 + 0.727i)T \)
13 \( 1 + (-0.973 + 0.230i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.973 + 0.230i)T \)
31 \( 1 + (0.0581 - 0.998i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.993 + 0.116i)T \)
53 \( 1 + (0.597 - 0.802i)T \)
59 \( 1 + (-0.396 + 0.918i)T \)
61 \( 1 + (0.835 - 0.549i)T \)
67 \( 1 + (0.396 - 0.918i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.973 + 0.230i)T \)
79 \( 1 + (0.993 - 0.116i)T \)
83 \( 1 + (0.893 + 0.448i)T \)
89 \( 1 + (-0.973 + 0.230i)T \)
97 \( 1 + (0.0581 + 0.998i)T \)
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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

−18.96465562732528797232481458140, −17.98661745343852713500800104698, −17.016942853320104931989570632401, −16.3247254689933204702463425690, −15.61720245653272347270814089166, −15.104745174638069222284660020571, −14.40238052853561151733205588975, −13.64831733422278056379789491584, −13.11500108610638152615617526580, −12.46865061588586371615628966863, −11.4311046837524379626307863041, −10.9049245256490248862275006032, −10.45982123625203276568439938909, −9.68082471381415362923765359029, −8.87604830388005676460283178954, −8.318031897205525111954706108515, −7.42269662133497033137472273528, −6.60155992322800848592236340219, −5.32542642404981829623312978900, −4.703254702015591154274903973953, −4.24646030911360269996668819236, −3.43909819084501634372117545468, −2.88713604499055916434374241992, −2.10291779887210455286140437939, −0.6415939672606209098753539293, 0.10163772898283026832400883124, 2.00139153755429092014028378504, 2.531003212138998717186325365909, 3.43687257822161382228945240957, 4.08096245173507909282801350534, 5.04144001880222563430892469498, 5.77598502425212442823884144103, 6.70420533718828889028681761936, 7.15871538382796905473847480875, 7.94654340957280687712274644194, 8.23094723333210065927227255141, 9.25463147600521279874333682195, 9.70588633902425078831502138848, 11.171281516908332244094374544433, 11.98948580200933936977368406939, 12.32904956280234490401573392330, 12.995322391780801332520380326464, 13.58773603706401398065447530456, 14.74736752708633775513790341699, 14.85062588345023733839302751122, 15.48918400391912299471094707928, 16.11777863719181416557445032867, 17.04392263605359825947456045255, 17.82905498454134935196365994857, 18.39436880746482300073087801083

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