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Properties

Label 4-380e2-1.1-c1e2-0-9
Degree $4$
Conductor $144400$
Sign $1$
Analytic cond. $9.20706$
Root an. cond. $1.74192$
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  − 4·3-s + 2·5-s − 4·7-s + 8·9-s − 4·11-s − 4·13-s − 8·15-s − 4·17-s − 2·19-s + 16·21-s − 12·23-s + 3·25-s − 12·27-s + 4·29-s − 8·31-s + 16·33-s − 8·35-s − 12·37-s + 16·39-s − 4·41-s + 4·43-s + 16·45-s − 4·47-s + 6·49-s + 16·51-s + 4·53-s − 8·55-s + ⋯
L(s)  = 1  − 2.30·3-s + 0.894·5-s − 1.51·7-s + 8/3·9-s − 1.20·11-s − 1.10·13-s − 2.06·15-s − 0.970·17-s − 0.458·19-s + 3.49·21-s − 2.50·23-s + 3/5·25-s − 2.30·27-s + 0.742·29-s − 1.43·31-s + 2.78·33-s − 1.35·35-s − 1.97·37-s + 2.56·39-s − 0.624·41-s + 0.609·43-s + 2.38·45-s − 0.583·47-s + 6/7·49-s + 2.24·51-s + 0.549·53-s − 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(144400\)    =    \(2^{4} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(9.20706\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 144400,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_k
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_m
17$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_be
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_ak
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cs
37$D_{4}$ \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_do
41$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cc
43$D_{4}$ \( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_de
47$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_dm
53$D_{4}$ \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_ci
59$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.59.aq_fu
61$D_{4}$ \( 1 + 16 T + 154 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.61.q_fy
67$D_{4}$ \( 1 + 4 T + 136 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_fg
71$D_{4}$ \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.71.aq_hq
73$D_{4}$ \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_eg
79$D_{4}$ \( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_fm
83$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_bq
89$D_{4}$ \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.89.ae_eg
97$D_{4}$ \( 1 + 12 T + 212 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.97.m_ie
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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

−10.94256066319059589801682795120, −10.68548639853506772334618426967, −10.16623922017043054936950871610, −10.13051823443032030003347563002, −9.537240522241727969188861774921, −9.087019736397755837878264379104, −8.306331735166356284250669934194, −7.69325377850132072651736083877, −6.99267862041749681055782655233, −6.61046132542093407854307811113, −6.32492184653728948339791911819, −5.78573202136722975021606241343, −5.26924804531103241711560797133, −5.17790341794470007537249143915, −4.27638772863881814162798045888, −3.60239909983863243422565985455, −2.51380771504828713069998884601, −1.95569357302930458987582739256, 0, 0, 1.95569357302930458987582739256, 2.51380771504828713069998884601, 3.60239909983863243422565985455, 4.27638772863881814162798045888, 5.17790341794470007537249143915, 5.26924804531103241711560797133, 5.78573202136722975021606241343, 6.32492184653728948339791911819, 6.61046132542093407854307811113, 6.99267862041749681055782655233, 7.69325377850132072651736083877, 8.306331735166356284250669934194, 9.087019736397755837878264379104, 9.537240522241727969188861774921, 10.13051823443032030003347563002, 10.16623922017043054936950871610, 10.68548639853506772334618426967, 10.94256066319059589801682795120

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