Properties

Label 4-3800e2-1.1-c1e2-0-4
Degree $4$
Conductor $14440000$
Sign $1$
Analytic cond. $920.706$
Root an. cond. $5.50846$
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  + 2·3-s − 2·7-s − 9-s − 2·13-s + 2·17-s − 2·19-s − 4·21-s + 2·23-s − 6·27-s + 2·29-s + 4·31-s + 16·37-s − 4·39-s − 4·43-s + 16·47-s − 9·49-s + 4·51-s + 2·53-s − 4·57-s + 26·59-s + 4·61-s + 2·63-s + 2·67-s + 4·69-s − 8·71-s + 22·73-s + 4·79-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s − 1/3·9-s − 0.554·13-s + 0.485·17-s − 0.458·19-s − 0.872·21-s + 0.417·23-s − 1.15·27-s + 0.371·29-s + 0.718·31-s + 2.63·37-s − 0.640·39-s − 0.609·43-s + 2.33·47-s − 9/7·49-s + 0.560·51-s + 0.274·53-s − 0.529·57-s + 3.38·59-s + 0.512·61-s + 0.251·63-s + 0.244·67-s + 0.481·69-s − 0.949·71-s + 2.57·73-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14440000\)    =    \(2^{6} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(920.706\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.620558373\)
\(L(\frac12)\) \(\approx\) \(3.620558373\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 64 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 16 T + 130 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 72 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 2 T + 99 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 26 T + 285 T^{2} - 26 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 124 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 85 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 22 T + 259 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T + 154 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−8.539900366826313680306669145625, −8.332111251559784136442004962044, −7.972442705255744215220339745387, −7.87002982830762091675834744919, −7.06691848521577099993701016655, −7.02278677780263789567725975061, −6.51099102239163279759406398975, −6.15643792310994844843355344565, −5.56494144360733950768964005829, −5.52775426131471740497381245548, −4.82329190583088995857440252842, −4.43599142038618552101781619289, −3.93003178651002150024846813935, −3.62039603597353601339392842058, −3.04142078804046083008289543764, −2.81811860622178336617157559684, −2.25577819245921500929562186542, −2.17304263108598225239215893536, −0.980856266321183460865432514058, −0.60763894153897535356229012661, 0.60763894153897535356229012661, 0.980856266321183460865432514058, 2.17304263108598225239215893536, 2.25577819245921500929562186542, 2.81811860622178336617157559684, 3.04142078804046083008289543764, 3.62039603597353601339392842058, 3.93003178651002150024846813935, 4.43599142038618552101781619289, 4.82329190583088995857440252842, 5.52775426131471740497381245548, 5.56494144360733950768964005829, 6.15643792310994844843355344565, 6.51099102239163279759406398975, 7.02278677780263789567725975061, 7.06691848521577099993701016655, 7.87002982830762091675834744919, 7.972442705255744215220339745387, 8.332111251559784136442004962044, 8.539900366826313680306669145625

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