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Properties

Label 1-4033-4033.2027-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.0648 + 0.997i$
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  + (0.939 + 0.342i)2-s + (0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (0.957 + 0.286i)5-s + (0.893 + 0.448i)6-s + (−0.686 − 0.727i)7-s + (0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (0.802 + 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.686 + 0.727i)12-s + (0.686 + 0.727i)13-s + (−0.396 − 0.918i)14-s + (0.918 + 0.396i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (0.957 + 0.286i)5-s + (0.893 + 0.448i)6-s + (−0.686 − 0.727i)7-s + (0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (0.802 + 0.597i)10-s + (−0.802 + 0.597i)11-s + (0.686 + 0.727i)12-s + (0.686 + 0.727i)13-s + (−0.396 − 0.918i)14-s + (0.918 + 0.396i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0648 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0648 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.0648 + 0.997i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.0648 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.730267753 + 3.980415176i\)
\(L(\frac12)\) \(\approx\) \(3.730267753 + 3.980415176i\)
\(L(1)\) \(\approx\) \(2.554423237 + 1.198664426i\)
\(L(1)\) \(\approx\) \(2.554423237 + 1.198664426i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (0.993 + 0.116i)T \)
5 \( 1 + (0.957 + 0.286i)T \)
7 \( 1 + (-0.686 - 0.727i)T \)
11 \( 1 + (-0.802 + 0.597i)T \)
13 \( 1 + (0.686 + 0.727i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.116 + 0.993i)T \)
31 \( 1 + (0.230 - 0.973i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (-0.998 - 0.0581i)T \)
53 \( 1 + (0.230 + 0.973i)T \)
59 \( 1 + (-0.396 + 0.918i)T \)
61 \( 1 + (0.549 + 0.835i)T \)
67 \( 1 + (0.957 - 0.286i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.597 - 0.802i)T \)
79 \( 1 + (0.686 - 0.727i)T \)
83 \( 1 + (0.993 + 0.116i)T \)
89 \( 1 + (-0.549 - 0.835i)T \)
97 \( 1 + (0.957 + 0.286i)T \)
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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

−18.49135046133113530345976581257, −17.978300394477363134057757029683, −16.5406630903190392593246665751, −15.996554912130940845662368413688, −15.56620545319018353217225479960, −14.655297133736727678422142945160, −14.05246154248945442490189966535, −13.48068838708715235960442559002, −12.95622390951457911332890021805, −12.49409271837631851089347375927, −11.62709573748070058603854152579, −10.57958836354475072509908910683, −9.837179588684109339338710149483, −9.64590165446950574752036840324, −8.37461330605733344691311431303, −8.07948058275651123532218113530, −6.73396561730639254755483221425, −6.237119699044999600777565821411, −5.47647938939239381791282324574, −4.90029041792568552286079375896, −3.677218099880481423117624578371, −3.16083374534137676987995331333, −2.463883053855381937622541790084, −1.884346257312034597290792958894, −0.83481150948388726869174533762, 1.536754002249010178356837449100, 2.13211695504478383291934365207, 2.91709309254105617144648783042, 3.649624098848724841452416875872, 4.27461850428744729068073263678, 5.10507736066085659768626409681, 6.07427836780931958404926387673, 6.68462653987674985858071709175, 7.28803330053835399700376038762, 8.032474052000415852898463128729, 8.91472669721651475022684570946, 9.66874445856219661881240206380, 10.45198210966514677192066563227, 10.84373882730586639486859773581, 12.096583465296755210067810497523, 12.98301108080577956460609002379, 13.33507568575919685522752807273, 13.73894067877034798254469189457, 14.496839097399204963381410371810, 15.09731486501669956860439099940, 15.76893413685193634148541811790, 16.421371532678963055574682251749, 17.1135888967077026249768398072, 17.92533711259319249517474767546, 18.669099749797247439557983949874

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