Properties

Label 4-5472e2-1.1-c1e2-0-9
Degree $4$
Conductor $29942784$
Sign $1$
Analytic cond. $1909.17$
Root an. cond. $6.61015$
Motivic weight $1$
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  + 5-s + 6·7-s + 3·11-s + 13-s + 8·17-s + 2·19-s − 7·23-s − 5·25-s + 3·29-s + 10·31-s + 6·35-s + 6·37-s − 8·41-s + 15·43-s − 5·47-s + 13·49-s + 13·53-s + 3·55-s + 13·59-s + 9·61-s + 65-s − 5·67-s − 8·71-s − 2·73-s + 18·77-s + 20·79-s + 10·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 2.26·7-s + 0.904·11-s + 0.277·13-s + 1.94·17-s + 0.458·19-s − 1.45·23-s − 25-s + 0.557·29-s + 1.79·31-s + 1.01·35-s + 0.986·37-s − 1.24·41-s + 2.28·43-s − 0.729·47-s + 13/7·49-s + 1.78·53-s + 0.404·55-s + 1.69·59-s + 1.15·61-s + 0.124·65-s − 0.610·67-s − 0.949·71-s − 0.234·73-s + 2.05·77-s + 2.25·79-s + 1.09·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29942784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29942784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29942784\)    =    \(2^{10} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1909.17\)
Root analytic conductor: \(6.61015\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5472} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29942784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.292760860\)
\(L(\frac12)\) \(\approx\) \(7.292760860\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 15 T + 138 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 13 T + 144 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 13 T + 156 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 9 T + 36 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 5 T + 136 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T + 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 10 T + 38 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−8.172817810352951082318929504919, −8.042375524290343568199281642834, −7.64163850794416004344796674669, −7.60684804537190771448330602739, −6.82532182922713467908780470717, −6.62081544760676749321757023340, −5.94312904475062751970603614611, −5.93744203980919319427482091975, −5.30073930920283809331793726000, −5.28049122798695779775084390241, −4.64820783072911342871159965307, −4.35730989303304622860553620190, −3.80242894078641353783412360705, −3.77855011782730182528548303337, −2.97834202402391071147252240709, −2.44895219823249263787187684678, −2.04333652287247103430469472446, −1.60667607946198771711868199936, −0.961262508287906583155104181122, −0.942066293094218888838118614618, 0.942066293094218888838118614618, 0.961262508287906583155104181122, 1.60667607946198771711868199936, 2.04333652287247103430469472446, 2.44895219823249263787187684678, 2.97834202402391071147252240709, 3.77855011782730182528548303337, 3.80242894078641353783412360705, 4.35730989303304622860553620190, 4.64820783072911342871159965307, 5.28049122798695779775084390241, 5.30073930920283809331793726000, 5.93744203980919319427482091975, 5.94312904475062751970603614611, 6.62081544760676749321757023340, 6.82532182922713467908780470717, 7.60684804537190771448330602739, 7.64163850794416004344796674669, 8.042375524290343568199281642834, 8.172817810352951082318929504919

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