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Properties

Label 4-3675e2-1.1-c1e2-0-16
Degree $4$
Conductor $13505625$
Sign $1$
Analytic cond. $861.130$
Root an. cond. $5.41710$
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 + 2·3-s − 2·4-s + 2·6-s − 3·8-s + 3·9-s − 4·12-s − 4·13-s + 16-s − 4·17-s + 3·18-s − 4·19-s + 4·23-s − 6·24-s − 4·26-s + 4·27-s − 6·29-s − 8·31-s + 2·32-s − 4·34-s − 6·36-s + 2·37-s − 4·38-s − 8·39-s − 4·43-s + 4·46-s − 12·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 4-s + 0.816·6-s − 1.06·8-s + 9-s − 1.15·12-s − 1.10·13-s + 1/4·16-s − 0.970·17-s + 0.707·18-s − 0.917·19-s + 0.834·23-s − 1.22·24-s − 0.784·26-s + 0.769·27-s − 1.11·29-s − 1.43·31-s + 0.353·32-s − 0.685·34-s − 36-s + 0.328·37-s − 0.648·38-s − 1.28·39-s − 0.609·43-s + 0.589·46-s − 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13505625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13505625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(861.130\)
Root analytic conductor: \(5.41710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13505625,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.2.ab_d
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.11.a_r
13$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_k
17$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_s
19$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_w
23$D_{4}$ \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bt
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$D_{4}$ \( 1 - 2 T + 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_cd
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.41.a_c
43$D_{4}$ \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_bt
47$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_eg
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_dy
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.61.y_kg
67$D_{4}$ \( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_ev
71$D_{4}$ \( 1 - 12 T + 173 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_gr
73$D_{4}$ \( 1 - 20 T + 226 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.73.au_is
79$D_{4}$ \( 1 - 8 T + 129 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_ez
83$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.83.q_fu
89$C_4$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_hm
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.97.ai_ic
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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

−8.313798991312808739980635862645, −7.976444359591331811775622725798, −7.70395045267224962821598704506, −7.39249907335176220872948892166, −6.69134345454870572587991957552, −6.67335702245178036284876233068, −6.12061581865831707299711789246, −5.55133369699384654514696678318, −5.05815482484268709892566769520, −4.89968887633151906875212811805, −4.45777957002355300150723667276, −4.18976262016750341872830435282, −3.67717147004736190513595586691, −3.28615716432559478867898721964, −2.94991247340893020985469757533, −2.34401500842170128687988530157, −1.86237780197871426945297222905, −1.42569959844573070098707952257, 0, 0, 1.42569959844573070098707952257, 1.86237780197871426945297222905, 2.34401500842170128687988530157, 2.94991247340893020985469757533, 3.28615716432559478867898721964, 3.67717147004736190513595586691, 4.18976262016750341872830435282, 4.45777957002355300150723667276, 4.89968887633151906875212811805, 5.05815482484268709892566769520, 5.55133369699384654514696678318, 6.12061581865831707299711789246, 6.67335702245178036284876233068, 6.69134345454870572587991957552, 7.39249907335176220872948892166, 7.70395045267224962821598704506, 7.976444359591331811775622725798, 8.313798991312808739980635862645

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