Properties

Label 4-1650e2-1.1-c1e2-0-12
Degree $4$
Conductor $2722500$
Sign $1$
Analytic cond. $173.588$
Root an. cond. $3.62978$
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  − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 4·7-s − 4·8-s + 3·9-s − 2·11-s − 6·12-s − 8·13-s + 8·14-s + 5·16-s − 6·18-s + 8·21-s + 4·22-s + 4·23-s + 8·24-s + 16·26-s − 4·27-s − 12·28-s − 4·29-s + 8·31-s − 6·32-s + 4·33-s + 9·36-s − 4·37-s + 16·39-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 1.51·7-s − 1.41·8-s + 9-s − 0.603·11-s − 1.73·12-s − 2.21·13-s + 2.13·14-s + 5/4·16-s − 1.41·18-s + 1.74·21-s + 0.852·22-s + 0.834·23-s + 1.63·24-s + 3.13·26-s − 0.769·27-s − 2.26·28-s − 0.742·29-s + 1.43·31-s − 1.06·32-s + 0.696·33-s + 3/2·36-s − 0.657·37-s + 2.56·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2722500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(173.588\)
Root analytic conductor: \(3.62978\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2722500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4303819742\)
\(L(\frac12)\) \(\approx\) \(0.4303819742\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 20 T + 192 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 56 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 128 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 20 T + 238 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−9.605628660020528236209559499732, −9.333332511103290808779608912193, −8.908545320898637105062975565319, −8.546891424508599762700157814556, −7.71814575527588000417211768187, −7.58971407525750955756796217795, −7.14752719066794291880042505207, −7.01511880992910579235640601944, −6.33067430583190903266392259833, −6.19756794711857585654899436434, −5.46461543246896391384646287503, −5.40983189353671470292244555015, −4.51289077712819823882741692532, −4.41633740331056703647907362450, −3.19775581989290473910105043033, −3.19282982815513290477477292791, −2.23313844318707728423525218137, −2.14640500837703947736986804191, −0.68879080601794403056645407291, −0.55946429140695181727147767153, 0.55946429140695181727147767153, 0.68879080601794403056645407291, 2.14640500837703947736986804191, 2.23313844318707728423525218137, 3.19282982815513290477477292791, 3.19775581989290473910105043033, 4.41633740331056703647907362450, 4.51289077712819823882741692532, 5.40983189353671470292244555015, 5.46461543246896391384646287503, 6.19756794711857585654899436434, 6.33067430583190903266392259833, 7.01511880992910579235640601944, 7.14752719066794291880042505207, 7.58971407525750955756796217795, 7.71814575527588000417211768187, 8.546891424508599762700157814556, 8.908545320898637105062975565319, 9.333332511103290808779608912193, 9.605628660020528236209559499732

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