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Properties

Label 4-1520e2-1.1-c1e2-0-22
Degree $4$
Conductor $2310400$
Sign $1$
Analytic cond. $147.313$
Root an. cond. $3.48385$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s − 8·7-s + 3·9-s − 2·11-s + 4·13-s + 2·15-s − 4·17-s − 7·19-s + 16·21-s + 6·23-s − 10·27-s − 5·29-s + 2·31-s + 4·33-s + 8·35-s − 4·37-s − 8·39-s + 6·41-s + 6·43-s − 3·45-s + 34·49-s + 8·51-s − 10·53-s + 2·55-s + 14·57-s + 3·59-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 3.02·7-s + 9-s − 0.603·11-s + 1.10·13-s + 0.516·15-s − 0.970·17-s − 1.60·19-s + 3.49·21-s + 1.25·23-s − 1.92·27-s − 0.928·29-s + 0.359·31-s + 0.696·33-s + 1.35·35-s − 0.657·37-s − 1.28·39-s + 0.937·41-s + 0.914·43-s − 0.447·45-s + 34/7·49-s + 1.12·51-s − 1.37·53-s + 0.269·55-s + 1.85·57-s + 0.390·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2310400\)    =    \(2^{8} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(147.313\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2310400,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
19$C_2$ \( 1 + 7 T + p T^{2} \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_ab
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.29.f_ae
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.31.ac_cl
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_ah
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.53.k_bv
59$C_2^2$ \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.59.ad_aby
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_abk
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2^2$ \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.71.n_du
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.73.ag_abl
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.79.al_bq
83$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.83.abc_ny
89$C_2^2$ \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.89.ah_abo
97$C_2^2$ \( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_acj
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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

−9.392813459486385073392930453541, −8.861817256456531708177074550795, −8.810807804624192130826012710379, −7.912008019633663019795855793100, −7.54933103379988826327758389596, −7.11604849509614428722895249577, −6.60733314705082890815393328489, −6.35779675985696740922609640070, −6.11183702704466892374436785412, −5.89929474360026346010270530637, −5.09008931674576238279297334464, −4.75308215230347394487865147496, −4.00517279255826453743088254989, −3.57104816470833818315872683285, −3.54070746346573287237937288943, −2.60487759655148434030595759890, −2.21538974500818549052233388950, −1.02783145871320189576730697162, 0, 0, 1.02783145871320189576730697162, 2.21538974500818549052233388950, 2.60487759655148434030595759890, 3.54070746346573287237937288943, 3.57104816470833818315872683285, 4.00517279255826453743088254989, 4.75308215230347394487865147496, 5.09008931674576238279297334464, 5.89929474360026346010270530637, 6.11183702704466892374436785412, 6.35779675985696740922609640070, 6.60733314705082890815393328489, 7.11604849509614428722895249577, 7.54933103379988826327758389596, 7.912008019633663019795855793100, 8.810807804624192130826012710379, 8.861817256456531708177074550795, 9.392813459486385073392930453541

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