Properties

Label 4-33e2-1.1-c5e2-0-0
Degree $4$
Conductor $1089$
Sign $1$
Analytic cond. $28.0123$
Root an. cond. $2.30057$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 18·3-s + 15·4-s − 38·5-s − 18·6-s − 18·7-s + 61·8-s + 243·9-s − 38·10-s + 242·11-s − 270·12-s − 66·13-s − 18·14-s + 684·15-s − 721·16-s − 920·17-s + 243·18-s − 2.93e3·19-s − 570·20-s + 324·21-s + 242·22-s + 5.24e3·23-s − 1.09e3·24-s + 2.65e3·25-s − 66·26-s − 2.91e3·27-s − 270·28-s + ⋯
L(s)  = 1  + 0.176·2-s − 1.15·3-s + 0.468·4-s − 0.679·5-s − 0.204·6-s − 0.138·7-s + 0.336·8-s + 9-s − 0.120·10-s + 0.603·11-s − 0.541·12-s − 0.108·13-s − 0.0245·14-s + 0.784·15-s − 0.704·16-s − 0.772·17-s + 0.176·18-s − 1.86·19-s − 0.318·20-s + 0.160·21-s + 0.106·22-s + 2.06·23-s − 0.389·24-s + 0.850·25-s − 0.0191·26-s − 0.769·27-s − 0.0650·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(28.0123\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{33} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1089,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.194348967\)
\(L(\frac12)\) \(\approx\) \(1.194348967\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_1$ \( ( 1 + p^{2} T )^{2} \)
11$C_1$ \( ( 1 - p^{2} T )^{2} \)
good2$D_{4}$ \( 1 - T - 7 p T^{2} - p^{5} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 + 38 T - 1214 T^{2} + 38 p^{5} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 18 T + 33382 T^{2} + 18 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 66 T - 135542 T^{2} + 66 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 920 T + 3050062 T^{2} + 920 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 2932 T + 7100102 T^{2} + 2932 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 5246 T + 19699918 T^{2} - 5246 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 12600 T + 79348870 T^{2} + 12600 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 9936 T + 53013118 T^{2} - 9936 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 5996 T + 139663118 T^{2} - 5996 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 24244 T + 337184038 T^{2} - 24244 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 20360 T + 277332086 T^{2} - 20360 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 5806 T + 419753950 T^{2} + 5806 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 40770 T + 1224329794 T^{2} - 40770 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 18212 T + 455377462 T^{2} - 18212 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 11398 T + 1131625826 T^{2} + 11398 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 65368 T + 3722340342 T^{2} - 65368 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 61446 T + 3799408318 T^{2} - 61446 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 53412 T + 4851340822 T^{2} - 53412 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 17122 T + 5872391094 T^{2} - 17122 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 14304 T + 6955271542 T^{2} + 14304 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 58140 T + 11297620726 T^{2} + 58140 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 183056 T + 25458744990 T^{2} + 183056 p^{5} T^{3} + p^{10} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.07809785890652776194182721632, −15.23431867127850081414487985713, −15.17492946747649701884565699229, −14.34243030527849428448463305240, −13.22545181873379446567013817213, −12.84285927985097752328433191834, −12.39609326600468949754693355121, −11.28812293790970773653276316684, −11.08882777826823977393015283963, −10.94095646036259825092598392044, −9.614806171845037458059377460363, −9.008339541023446007286035294706, −7.987752016260584808050827269045, −6.88320511046792337825395795697, −6.76955182103526619177115805629, −5.73734338741906870614759001660, −4.60762522429610294231293191605, −4.06079175080597074273269044081, −2.35251158114376928065764876110, −0.70022603535538289271875179639, 0.70022603535538289271875179639, 2.35251158114376928065764876110, 4.06079175080597074273269044081, 4.60762522429610294231293191605, 5.73734338741906870614759001660, 6.76955182103526619177115805629, 6.88320511046792337825395795697, 7.987752016260584808050827269045, 9.008339541023446007286035294706, 9.614806171845037458059377460363, 10.94095646036259825092598392044, 11.08882777826823977393015283963, 11.28812293790970773653276316684, 12.39609326600468949754693355121, 12.84285927985097752328433191834, 13.22545181873379446567013817213, 14.34243030527849428448463305240, 15.17492946747649701884565699229, 15.23431867127850081414487985713, 16.07809785890652776194182721632

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