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Properties

Label 4-2376e2-1.1-c1e2-0-25
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·11-s − 4·17-s − 6·23-s − 10·25-s − 8·29-s + 8·31-s − 6·37-s − 4·41-s + 4·43-s − 2·47-s − 7·49-s − 8·53-s − 14·59-s + 16·67-s − 8·71-s − 8·73-s − 8·79-s − 16·83-s − 16·89-s − 6·97-s − 16·101-s + 8·103-s − 4·109-s − 16·113-s + 3·121-s + 127-s + 131-s + ⋯
L(s)  = 1  − 0.603·11-s − 0.970·17-s − 1.25·23-s − 2·25-s − 1.48·29-s + 1.43·31-s − 0.986·37-s − 0.624·41-s + 0.609·43-s − 0.291·47-s − 49-s − 1.09·53-s − 1.82·59-s + 1.95·67-s − 0.949·71-s − 0.936·73-s − 0.900·79-s − 1.75·83-s − 1.69·89-s − 0.609·97-s − 1.59·101-s + 0.788·103-s − 0.383·109-s − 1.50·113-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.7.a_h
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$D_{4}$ \( 1 + 4 T + 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_bf
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$D_{4}$ \( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_bb
29$D_{4}$ \( 1 + 8 T + 67 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_cp
31$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_by
37$D_{4}$ \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_cd
41$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_x
43$D_{4}$ \( 1 - 4 T + 83 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_df
47$D_{4}$ \( 1 + 2 T + 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_cp
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.53.i_es
59$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.59.o_gl
61$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.61.a_k
67$D_{4}$ \( 1 - 16 T + 170 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.67.aq_go
71$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_bu
73$D_{4}$ \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_by
79$D_{4}$ \( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_eh
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.83.q_iw
89$D_{4}$ \( 1 + 16 T + 214 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_ig
97$D_{4}$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.97.g_dn
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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

−8.581990946359635586186070129429, −8.425861682688599546074730984999, −7.957376825495988636459786094571, −7.81657217232928436309713900688, −7.15493238770020136798806266308, −6.99886003603691598967698941032, −6.30929967855902857287597877146, −6.13423503666358604664728391549, −5.52677564769726307032242078145, −5.46811064704088896741403166610, −4.64404877437031747704971241090, −4.37457842409541625313040805463, −4.00293648029006650730013610631, −3.44180649874906602485806169891, −2.93875949412210388573819131003, −2.41117014053137635668286723758, −1.78656025314659326333171156104, −1.52936121797010545390920367117, 0, 0, 1.52936121797010545390920367117, 1.78656025314659326333171156104, 2.41117014053137635668286723758, 2.93875949412210388573819131003, 3.44180649874906602485806169891, 4.00293648029006650730013610631, 4.37457842409541625313040805463, 4.64404877437031747704971241090, 5.46811064704088896741403166610, 5.52677564769726307032242078145, 6.13423503666358604664728391549, 6.30929967855902857287597877146, 6.99886003603691598967698941032, 7.15493238770020136798806266308, 7.81657217232928436309713900688, 7.957376825495988636459786094571, 8.425861682688599546074730984999, 8.581990946359635586186070129429

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