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Properties

Label 1-4033-4033.1533-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.547 + 0.837i$
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  + (0.173 + 0.984i)2-s + (0.597 + 0.802i)3-s + (−0.939 + 0.342i)4-s + (−0.286 − 0.957i)5-s + (−0.686 + 0.727i)6-s + (−0.686 + 0.727i)7-s + (−0.5 − 0.866i)8-s + (−0.286 + 0.957i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.835 − 0.549i)12-s + (0.973 + 0.230i)13-s + (−0.835 − 0.549i)14-s + (0.597 − 0.802i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.597 + 0.802i)3-s + (−0.939 + 0.342i)4-s + (−0.286 − 0.957i)5-s + (−0.686 + 0.727i)6-s + (−0.686 + 0.727i)7-s + (−0.5 − 0.866i)8-s + (−0.286 + 0.957i)9-s + (0.893 − 0.448i)10-s + (0.893 + 0.448i)11-s + (−0.835 − 0.549i)12-s + (0.973 + 0.230i)13-s + (−0.835 − 0.549i)14-s + (0.597 − 0.802i)15-s + (0.766 − 0.642i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.547 + 0.837i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.547 + 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003439467 + 1.854582363i\)
\(L(\frac12)\) \(\approx\) \(1.003439467 + 1.854582363i\)
\(L(1)\) \(\approx\) \(0.9443595986 + 0.8630019088i\)
\(L(1)\) \(\approx\) \(0.9443595986 + 0.8630019088i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 + (0.597 + 0.802i)T \)
5 \( 1 + (-0.286 - 0.957i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.893 + 0.448i)T \)
13 \( 1 + (0.973 + 0.230i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.893 - 0.448i)T \)
31 \( 1 + (0.396 + 0.918i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.973 - 0.230i)T \)
53 \( 1 + (-0.835 - 0.549i)T \)
59 \( 1 + (0.597 - 0.802i)T \)
61 \( 1 + (0.893 - 0.448i)T \)
67 \( 1 + (-0.0581 - 0.998i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.0581 - 0.998i)T \)
79 \( 1 + (0.893 + 0.448i)T \)
83 \( 1 + (-0.286 - 0.957i)T \)
89 \( 1 + (0.396 - 0.918i)T \)
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

−18.61534174894505903677210886049, −17.79133446980814284695656684688, −17.20459433626803143066820757952, −16.156939192197365561064766778358, −15.28139826922771333132121666481, −14.40388653741904313204756508440, −14.04333752046849374705153357191, −13.48348425113693139093414891580, −12.82440232523047100683748529894, −12.02528390897081078784042598059, −11.4003249538133185718548563863, −10.79823098042498896748095157524, −9.98655263402186903872512319280, −9.320982377483288555659301065929, −8.5900956297085328425352443263, −7.78168828606266390892799007556, −7.01504792867473196276909765435, −6.304179201494611592904374962547, −5.69838427863000492813326696390, −4.060540248161488993612310711730, −3.82971638912791848405807427398, −2.97835358858660638274310547480, −2.553211383540942409293811939256, −1.26225253672156589119020449741, −0.795165630981485258252443932419, 0.81370489181467052574627740718, 2.09276151459689571494362327503, 3.257644918165660300991842205643, 3.86760482187707025719608014603, 4.55582513067993965425082807465, 5.0983093709320235909002564126, 6.126221086167917996610919125231, 6.55382334497231487639958988119, 7.6919902068978709236318566728, 8.52678808300011965468969245789, 8.87057228345950461582386163702, 9.2802418280251826702697118154, 10.12041476945577128878674473892, 11.08947546155757085039057724577, 12.15652435821667358255686018249, 12.709487060408561673632691671772, 13.3170902371785516174340449281, 14.11158665310752066521966252629, 14.8029411569856559512871857582, 15.48206217726668396556461476084, 15.86691700153474393758018711812, 16.42766537050580656462435445212, 17.16808133835091539849361865995, 17.64424936330276850249197853495, 18.91223003847034562266206725094

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