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Properties

Label 4-2376e2-1.1-c1e2-0-24
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  − 2·7-s − 2·11-s − 2·13-s + 4·17-s − 2·19-s − 8·25-s + 4·29-s − 4·31-s − 2·37-s − 12·43-s − 8·47-s − 9·49-s + 8·53-s − 8·59-s − 18·61-s − 6·67-s − 8·71-s + 2·73-s + 4·77-s − 14·79-s − 8·83-s + 8·89-s + 4·91-s − 18·97-s + 20·101-s − 6·103-s − 20·107-s + ⋯
L(s)  = 1  − 0.755·7-s − 0.603·11-s − 0.554·13-s + 0.970·17-s − 0.458·19-s − 8/5·25-s + 0.742·29-s − 0.718·31-s − 0.328·37-s − 1.82·43-s − 1.16·47-s − 9/7·49-s + 1.09·53-s − 1.04·59-s − 2.30·61-s − 0.733·67-s − 0.949·71-s + 0.234·73-s + 0.455·77-s − 1.57·79-s − 0.878·83-s + 0.847·89-s + 0.419·91-s − 1.82·97-s + 1.99·101-s − 0.591·103-s − 1.93·107-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$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_n
13$D_{4}$ \( 1 + 2 T + 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_z
17$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_u
19$D_{4}$ \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_v
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$D_{4}$ \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_ci
31$D_{4}$ \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_bi
37$D_{4}$ \( 1 + 2 T + 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_cp
41$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.41.a_cw
43$D_{4}$ \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dm
47$D_{4}$ \( 1 + 8 T + 92 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_do
53$D_{4}$ \( 1 - 8 T + 24 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_y
59$D_{4}$ \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_fc
61$D_{4}$ \( 1 + 18 T + 201 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.61.s_ht
67$D_{4}$ \( 1 + 6 T + 15 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_p
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$D_{4}$ \( 1 - 2 T + 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.73.ac_bx
79$D_{4}$ \( 1 + 14 T + 157 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_gb
83$D_{4}$ \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.83.i_fu
89$D_{4}$ \( 1 - 8 T + 176 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_gu
97$D_{4}$ \( 1 + 18 T + 243 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.97.s_jj
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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

−8.764927460730068954400118634864, −8.284487099304679474942791076572, −8.013511916067550515911286670727, −7.73879756813616101118742613999, −7.10335913968885133585234178187, −7.04064346997443671924927510112, −6.35255370174172664429436387223, −6.07823473370452348454850611790, −5.70769728577183796210943621695, −5.26664865308619342742685209578, −4.67535876575677882375754483115, −4.57275416916763711201910048391, −3.61895211959571737814884818952, −3.60076607435111630219229591335, −2.82354590112778779847629857859, −2.69413826853995303157502559020, −1.66071722704919012519710257283, −1.52537445965912890582881174223, 0, 0, 1.52537445965912890582881174223, 1.66071722704919012519710257283, 2.69413826853995303157502559020, 2.82354590112778779847629857859, 3.60076607435111630219229591335, 3.61895211959571737814884818952, 4.57275416916763711201910048391, 4.67535876575677882375754483115, 5.26664865308619342742685209578, 5.70769728577183796210943621695, 6.07823473370452348454850611790, 6.35255370174172664429436387223, 7.04064346997443671924927510112, 7.10335913968885133585234178187, 7.73879756813616101118742613999, 8.013511916067550515911286670727, 8.284487099304679474942791076572, 8.764927460730068954400118634864

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