Properties

Label 4-29e2-1.1-c1e2-0-0
Degree $4$
Conductor $841$
Sign $1$
Analytic cond. $0.0536228$
Root an. cond. $0.481213$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·3-s + 4-s − 2·5-s − 4·6-s − 9-s + 4·10-s + 2·11-s + 2·12-s − 2·13-s − 4·15-s + 16-s − 4·17-s + 2·18-s + 12·19-s − 2·20-s − 4·22-s − 4·23-s − 7·25-s + 4·26-s − 6·27-s + 2·29-s + 8·30-s + 6·31-s + 2·32-s + 4·33-s + 8·34-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s − 1/3·9-s + 1.26·10-s + 0.603·11-s + 0.577·12-s − 0.554·13-s − 1.03·15-s + 1/4·16-s − 0.970·17-s + 0.471·18-s + 2.75·19-s − 0.447·20-s − 0.852·22-s − 0.834·23-s − 7/5·25-s + 0.784·26-s − 1.15·27-s + 0.371·29-s + 1.46·30-s + 1.07·31-s + 0.353·32-s + 0.696·33-s + 1.37·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $1$
Analytic conductor: \(0.0536228\)
Root analytic conductor: \(0.481213\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 841,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2915215656\)
\(L(\frac12)\) \(\approx\) \(0.2915215656\)
\(L(\frac{3}{2})\) 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)$
bad29$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 21 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T + 77 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 35 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$C_4$ \( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 2 T + 157 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−17.58649780159004962164621193908, −17.31343986241515149330655261806, −16.05888388043654018981880321944, −15.93806773944902668096847404058, −15.23224358305790267581443761926, −14.45200033040581709499805955154, −13.81654455481717156717333616108, −13.66732976667500941952922868953, −12.08232866650066832713858687587, −11.93782397100020205133590566736, −11.16889974274860287266489808404, −10.04252502469782990793395660236, −9.347008880241771359683944351195, −9.184733495052158934327725309962, −8.111694777874299421477288589426, −8.013350533114057240909833751006, −7.11407190598914430481785081554, −5.71193103759604998159133555315, −4.12707082881298541650894475106, −2.92477068592233051723034627885, 2.92477068592233051723034627885, 4.12707082881298541650894475106, 5.71193103759604998159133555315, 7.11407190598914430481785081554, 8.013350533114057240909833751006, 8.111694777874299421477288589426, 9.184733495052158934327725309962, 9.347008880241771359683944351195, 10.04252502469782990793395660236, 11.16889974274860287266489808404, 11.93782397100020205133590566736, 12.08232866650066832713858687587, 13.66732976667500941952922868953, 13.81654455481717156717333616108, 14.45200033040581709499805955154, 15.23224358305790267581443761926, 15.93806773944902668096847404058, 16.05888388043654018981880321944, 17.31343986241515149330655261806, 17.58649780159004962164621193908

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