Properties

Label 4-2240e2-1.1-c1e2-0-10
Degree $4$
Conductor $5017600$
Sign $1$
Analytic cond. $319.926$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + 2·7-s − 9-s + 11-s − 13-s + 2·15-s + 11·17-s + 6·19-s − 2·21-s − 2·23-s + 3·25-s − 5·29-s − 4·31-s − 33-s − 4·35-s + 39-s + 6·41-s + 6·43-s + 2·45-s + 9·47-s + 3·49-s − 11·51-s + 18·53-s − 2·55-s − 6·57-s + 8·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 0.755·7-s − 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.516·15-s + 2.66·17-s + 1.37·19-s − 0.436·21-s − 0.417·23-s + 3/5·25-s − 0.928·29-s − 0.718·31-s − 0.174·33-s − 0.676·35-s + 0.160·39-s + 0.937·41-s + 0.914·43-s + 0.298·45-s + 1.31·47-s + 3/7·49-s − 1.54·51-s + 2.47·53-s − 0.269·55-s − 0.794·57-s + 1.04·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5017600\)    =    \(2^{12} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(319.926\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5017600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.280305656\)
\(L(\frac12)\) \(\approx\) \(2.280305656\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 9 T + 110 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 22 T + 226 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 11 T + 150 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 15 T + 212 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
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

−9.033380050978804627668454502075, −8.972134148734138360101930526894, −8.274250093768992320060826550078, −8.037678349064237585599297107016, −7.66276431483655092380456668445, −7.29747360381492402139734859236, −7.14688642410786182131388040064, −6.61003106406653157611852128796, −5.70184380686824654670719035177, −5.60217909648394272337989738456, −5.41397203610325815533621676106, −5.14687666048628455347750689563, −4.05638536675559219564201833078, −4.05567093416085006485325311950, −3.70745205555007336879750470996, −2.96326989046568118604550731557, −2.57566179652403389699986079062, −1.79022099605831759545177029990, −0.944072251671515634703874090699, −0.74305058253375864616175542856, 0.74305058253375864616175542856, 0.944072251671515634703874090699, 1.79022099605831759545177029990, 2.57566179652403389699986079062, 2.96326989046568118604550731557, 3.70745205555007336879750470996, 4.05567093416085006485325311950, 4.05638536675559219564201833078, 5.14687666048628455347750689563, 5.41397203610325815533621676106, 5.60217909648394272337989738456, 5.70184380686824654670719035177, 6.61003106406653157611852128796, 7.14688642410786182131388040064, 7.29747360381492402139734859236, 7.66276431483655092380456668445, 8.037678349064237585599297107016, 8.274250093768992320060826550078, 8.972134148734138360101930526894, 9.033380050978804627668454502075

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