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Properties

Label 1-4033-4033.1089-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.449 - 0.893i$
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 + (0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s − 6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.766 + 0.642i)12-s + (−0.173 − 0.984i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s − 6-s + (−0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.766 + 0.642i)12-s + (−0.173 − 0.984i)13-s + (0.5 + 0.866i)14-s + (−0.939 + 0.342i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.449 - 0.893i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.449 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5632709887 - 0.9136421686i\)
\(L(\frac12)\) \(\approx\) \(0.5632709887 - 0.9136421686i\)
\(L(1)\) \(\approx\) \(0.6241473766 - 0.4474137832i\)
\(L(1)\) \(\approx\) \(0.6241473766 - 0.4474137832i\)

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 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−18.87404124216841547676647302138, −18.19053723961205806175312452925, −16.99914975718368777747472717736, −16.62497936618557897989887359249, −15.837692224971141109191455305758, −15.36785421848798205129613284171, −14.93312271973875314564862358208, −14.21221882811179044882127998763, −13.50423857771225065601356193632, −12.48287157977234674600264493948, −11.60650949944036046609600944492, −10.99207723025228830103995285985, −10.0735460985483732227509425046, −9.56337007606317351183071013030, −8.9780296351788710188932704823, −8.39654180410087719843504413897, −7.50747772812428691478289273565, −6.83270171083719467135752415901, −6.48515711532777815194331564824, −5.09788373342295583319940013843, −4.436875706853635052363424897, −3.82130069638670958779154734045, −2.516329043185290800867636114067, −2.32685979318428525233525811457, −0.6796129254473068299206594903, 0.61021857450006401061099561087, 1.15642195841373024474992554082, 2.272130321375776043881104187, 3.20921858515277907120346699234, 3.620489637354565798956820006, 4.15221310145558276936745359958, 5.71693332940755526644789684158, 6.62587483363437519008157104616, 7.28244670114515409163142366498, 7.962862576814455628391465713804, 8.556832944968394239515909686937, 9.02194143379120044024167841140, 9.89105452376514117256418819567, 10.658807568448620094989359391711, 11.3176131694059788204659020125, 12.34016042651265751705433338651, 12.578231343340967621809299753814, 13.199818620658878244321721432907, 13.944800121634083381602091533602, 15.02108123444712583574128642035, 15.565015516259779566766444331841, 16.33441664370852363723389541994, 17.04961346433098182932741652937, 17.56631776845657725637979016608, 18.609405605873043392382579165625

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