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Properties

Label 1-4033-4033.104-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.856 + 0.516i$
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  + 2-s + (−0.286 − 0.957i)3-s + 4-s + (−0.893 + 0.448i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s + 8-s + (−0.835 + 0.549i)9-s + (−0.893 + 0.448i)10-s + (−0.893 − 0.448i)11-s + (−0.286 − 0.957i)12-s + (0.893 − 0.448i)13-s + (−0.835 − 0.549i)14-s + (0.686 + 0.727i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.286 − 0.957i)3-s + 4-s + (−0.893 + 0.448i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s + 8-s + (−0.835 + 0.549i)9-s + (−0.893 + 0.448i)10-s + (−0.893 − 0.448i)11-s + (−0.286 − 0.957i)12-s + (0.893 − 0.448i)13-s + (−0.835 − 0.549i)14-s + (0.686 + 0.727i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.856 + 0.516i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.856 + 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2286053108 - 0.8221977912i\)
\(L(\frac12)\) \(\approx\) \(-0.2286053108 - 0.8221977912i\)
\(L(1)\) \(\approx\) \(1.054341595 - 0.5307012829i\)
\(L(1)\) \(\approx\) \(1.054341595 - 0.5307012829i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.286 - 0.957i)T \)
5 \( 1 + (-0.893 + 0.448i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (-0.893 - 0.448i)T \)
13 \( 1 + (0.893 - 0.448i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.835 + 0.549i)T \)
31 \( 1 + (-0.597 + 0.802i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (0.286 - 0.957i)T \)
53 \( 1 + (-0.973 - 0.230i)T \)
59 \( 1 + (-0.686 - 0.727i)T \)
61 \( 1 + (-0.396 + 0.918i)T \)
67 \( 1 + (0.286 - 0.957i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (-0.686 - 0.727i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (0.0581 - 0.998i)T \)
97 \( 1 + (-0.597 - 0.802i)T \)
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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

−19.05999190082551447461280604649, −18.227554705387660496623767098965, −17.205890963624631362810472992403, −16.30532669248178189619382137615, −16.02533090467892806880544096607, −15.54996062229783830342405363323, −15.047815526893284539788112654813, −14.16961483065528670882598891463, −13.31287341447227474960705728873, −12.61474962070924266604285928737, −12.10580706799419805986420862410, −11.37633947998500896048740946754, −10.839674259344854013580994304713, −9.9714528540111361100199581476, −9.3403584010841170915483425747, −8.28236174175684331137686266741, −7.792639061321055070966604520878, −6.61869829858684637066980966032, −5.93973776266200388397744165667, −5.4454463964626923592792518155, −4.42032936136930355589191413777, −4.07803495424527423347910582969, −3.31448219533063139115900854551, −2.62297748018390560475257958362, −1.42663992986271528934855288655, 0.18082850360306000997606784182, 1.08976733452429358244689077689, 2.38107713940926746412704528202, 3.08090812270667895525537044876, 3.496488201416678495867368203294, 4.58454823043492869027534007887, 5.37094204946362159217249417658, 6.16299776182644312128096394723, 6.824707868063508313433502829244, 7.30309221027064469077181767255, 7.963467018845257829082576038783, 8.74879021114578981666118176250, 10.18963235670868869839333064454, 10.82981740103867407977922683449, 11.25922086193128540397790562361, 12.082741285843641813426817501789, 12.62283543292235589693782747041, 13.3590077126768705421758670650, 13.79448989178563527731197859271, 14.37088356463355587839472741736, 15.560963984313780044535585811558, 16.00536499852102273871893356966, 16.27040494595466807498970565513, 17.4120168415052320092982866458, 18.36560393370135710093711172911

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