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Properties

Label 4-2106e2-1.1-c1e2-0-36
Degree $4$
Conductor $4435236$
Sign $1$
Analytic cond. $282.794$
Root an. cond. $4.10079$
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  + 2·2-s + 3·4-s − 2·5-s − 4·7-s + 4·8-s − 4·10-s + 2·11-s − 2·13-s − 8·14-s + 5·16-s − 6·17-s − 6·19-s − 6·20-s + 4·22-s − 2·23-s − 4·25-s − 4·26-s − 12·28-s − 14·29-s − 10·31-s + 6·32-s − 12·34-s + 8·35-s − 12·38-s − 8·40-s + 12·41-s − 4·43-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.51·7-s + 1.41·8-s − 1.26·10-s + 0.603·11-s − 0.554·13-s − 2.13·14-s + 5/4·16-s − 1.45·17-s − 1.37·19-s − 1.34·20-s + 0.852·22-s − 0.417·23-s − 4/5·25-s − 0.784·26-s − 2.26·28-s − 2.59·29-s − 1.79·31-s + 1.06·32-s − 2.05·34-s + 1.35·35-s − 1.94·38-s − 1.26·40-s + 1.87·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4435236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4435236\)    =    \(2^{2} \cdot 3^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(282.794\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4435236,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_i
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$D_{4}$ \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_l
17$D_{4}$ \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bf
19$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bj
23$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_u
29$D_{4}$ \( 1 + 14 T + 104 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.29.o_ea
31$D_{4}$ \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_ci
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$D_{4}$ \( 1 - 12 T + 115 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_el
43$D_{4}$ \( 1 + 4 T + 87 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_dj
47$D_{4}$ \( 1 - 14 T + 116 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.47.ao_em
53$D_{4}$ \( 1 + 6 T + 112 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_ei
59$D_{4}$ \( 1 + 2 T + 71 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.59.c_ct
61$D_{4}$ \( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_eq
67$D_{4}$ \( 1 + 10 T + 111 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_eh
71$D_{4}$ \( 1 - 2 T + 68 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.71.ac_cq
73$D_{4}$ \( 1 + 8 T + 87 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_dj
79$D_{4}$ \( 1 + 26 T + 324 T^{2} + 26 p T^{3} + p^{2} T^{4} \) 2.79.ba_mm
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.89.a_fa
97$D_{4}$ \( 1 + 8 T + 183 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hb
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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

−8.784934850101165934246843827449, −8.774447140447442094980009399879, −7.80287411903462176417688021316, −7.56282886959078876005975505209, −7.28243021840049202166036840955, −6.93485003499532431410455026763, −6.29061731250863384203908257769, −6.26714400565631409791707907173, −5.60475574343432639877480507577, −5.56776314821544267863614170885, −4.60207658478157823872049634464, −4.29077966791000947212632141673, −3.87846455046778053668454235375, −3.86069195378194905468457594011, −3.14211970387558329878235051603, −2.69995911268929244384192979799, −2.02896404376657247281646781007, −1.72747995305314722561373208036, 0, 0, 1.72747995305314722561373208036, 2.02896404376657247281646781007, 2.69995911268929244384192979799, 3.14211970387558329878235051603, 3.86069195378194905468457594011, 3.87846455046778053668454235375, 4.29077966791000947212632141673, 4.60207658478157823872049634464, 5.56776314821544267863614170885, 5.60475574343432639877480507577, 6.26714400565631409791707907173, 6.29061731250863384203908257769, 6.93485003499532431410455026763, 7.28243021840049202166036840955, 7.56282886959078876005975505209, 7.80287411903462176417688021316, 8.774447140447442094980009399879, 8.784934850101165934246843827449

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