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Properties

Label 4-1575e2-1.1-c1e2-0-12
Degree $4$
Conductor $2480625$
Sign $1$
Analytic cond. $158.166$
Root an. cond. $3.54632$
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-s − 2·4-s − 2·7-s − 3·8-s − 4·11-s + 2·13-s − 2·14-s + 16-s − 4·17-s − 4·22-s + 8·23-s + 2·26-s + 4·28-s − 10·29-s − 6·31-s + 2·32-s − 4·34-s − 6·37-s − 14·41-s − 8·43-s + 8·44-s + 8·46-s − 4·47-s + 3·49-s − 4·52-s + 8·53-s + 6·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s − 0.755·7-s − 1.06·8-s − 1.20·11-s + 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s − 0.852·22-s + 1.66·23-s + 0.392·26-s + 0.755·28-s − 1.85·29-s − 1.07·31-s + 0.353·32-s − 0.685·34-s − 0.986·37-s − 2.18·41-s − 1.21·43-s + 1.20·44-s + 1.17·46-s − 0.583·47-s + 3/7·49-s − 0.554·52-s + 1.09·53-s + 0.801·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2480625\)    =    \(3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(158.166\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2480625,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
11$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_v
13$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_w
17$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_s
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$D_{4}$ \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_cf
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.29.k_df
31$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_ba
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$D_{4}$ \( 1 + 14 T + 126 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.41.o_ew
43$D_{4}$ \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_dt
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.47.e_du
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_dy
59$D_{4}$ \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_du
61$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_di
67$D_{4}$ \( 1 - 4 T + 133 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_fd
71$D_{4}$ \( 1 + 4 T + 101 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_dx
73$D_{4}$ \( 1 - 22 T + 262 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.73.aw_kc
79$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \) 2.79.a_bh
83$D_{4}$ \( 1 + 2 T + 122 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.83.c_es
89$D_{4}$ \( 1 + 30 T + 398 T^{2} + 30 p T^{3} + p^{2} T^{4} \) 2.89.be_pi
97$D_{4}$ \( 1 + 6 T + 198 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_hq
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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

−9.016674763939395922533721064260, −9.015818767480641590494540269519, −8.414233578723368141617527311173, −8.283868080415678386542766332755, −7.53148923264926700975358271651, −7.15581737363023139572388353792, −6.75009883347375055028335954907, −6.44572772153427345416446092618, −5.66712829094775006233928476008, −5.42538953373525037613747107251, −5.04280082945156508818286961776, −4.84122488701842089366980507762, −3.96703907466446787406158731692, −3.83932341101615808283304805012, −3.22755375148447618726290811510, −2.93410291197492932117551952451, −2.09151690798107021125714979894, −1.44631677378036851450710791691, 0, 0, 1.44631677378036851450710791691, 2.09151690798107021125714979894, 2.93410291197492932117551952451, 3.22755375148447618726290811510, 3.83932341101615808283304805012, 3.96703907466446787406158731692, 4.84122488701842089366980507762, 5.04280082945156508818286961776, 5.42538953373525037613747107251, 5.66712829094775006233928476008, 6.44572772153427345416446092618, 6.75009883347375055028335954907, 7.15581737363023139572388353792, 7.53148923264926700975358271651, 8.283868080415678386542766332755, 8.414233578723368141617527311173, 9.015818767480641590494540269519, 9.016674763939395922533721064260

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