Properties

Label 4-7600e2-1.1-c1e2-0-8
Degree $4$
Conductor $57760000$
Sign $1$
Analytic cond. $3682.82$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 4·7-s + 8·9-s + 4·11-s + 4·13-s + 4·17-s + 2·19-s + 16·21-s − 12·23-s − 12·27-s + 4·29-s + 8·31-s − 16·33-s + 12·37-s − 16·39-s − 4·41-s + 4·43-s − 4·47-s + 6·49-s − 16·51-s − 4·53-s − 8·57-s − 16·59-s − 16·61-s − 32·63-s − 4·67-s + 48·69-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.51·7-s + 8/3·9-s + 1.20·11-s + 1.10·13-s + 0.970·17-s + 0.458·19-s + 3.49·21-s − 2.50·23-s − 2.30·27-s + 0.742·29-s + 1.43·31-s − 2.78·33-s + 1.97·37-s − 2.56·39-s − 0.624·41-s + 0.609·43-s − 0.583·47-s + 6/7·49-s − 2.24·51-s − 0.549·53-s − 1.05·57-s − 2.08·59-s − 2.04·61-s − 4.03·63-s − 0.488·67-s + 5.77·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(57760000\)    =    \(2^{8} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(3682.82\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 57760000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 154 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 136 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 212 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.47579100081873647343737123183, −7.43960140065000642912778837680, −6.58918775283060984723165495386, −6.40097313956949129748397024172, −6.20981758139523704931577330373, −6.17512027789531467168869845890, −5.68859993282908055755918509507, −5.65973773064232195339946827670, −4.72608903831480755906828927504, −4.64687975272024600952456563725, −4.27312918786731683231944592029, −3.79512559058064206018745108175, −3.40709066735849313178600114329, −3.07811412797321552844172328627, −2.53650530743988428683525451315, −1.70572480122082180628276553013, −1.19998502955432035098250849677, −1.00071086399922624652836442145, 0, 0, 1.00071086399922624652836442145, 1.19998502955432035098250849677, 1.70572480122082180628276553013, 2.53650530743988428683525451315, 3.07811412797321552844172328627, 3.40709066735849313178600114329, 3.79512559058064206018745108175, 4.27312918786731683231944592029, 4.64687975272024600952456563725, 4.72608903831480755906828927504, 5.65973773064232195339946827670, 5.68859993282908055755918509507, 6.17512027789531467168869845890, 6.20981758139523704931577330373, 6.40097313956949129748397024172, 6.58918775283060984723165495386, 7.43960140065000642912778837680, 7.47579100081873647343737123183

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