Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4·4-s − 6·5-s − 3·9-s + 8·12-s + 12·15-s + 12·16-s + 24·20-s − 18·23-s + 17·25-s + 14·27-s − 10·31-s + 12·36-s + 14·37-s + 18·45-s − 24·47-s − 24·48-s − 14·49-s + 12·53-s − 30·59-s − 48·60-s − 32·64-s + 26·67-s + 36·69-s − 6·71-s − 34·75-s − 72·80-s + ⋯
L(s)  = 1  − 1.15·3-s − 2·4-s − 2.68·5-s − 9-s + 2.30·12-s + 3.09·15-s + 3·16-s + 5.36·20-s − 3.75·23-s + 17/5·25-s + 2.69·27-s − 1.79·31-s + 2·36-s + 2.30·37-s + 2.68·45-s − 3.50·47-s − 3.46·48-s − 2·49-s + 1.64·53-s − 3.90·59-s − 6.19·60-s − 4·64-s + 3.17·67-s + 4.33·69-s − 0.712·71-s − 3.92·75-s − 8.04·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14641\)    =    \(11^{4}\)
Sign: $1$
Analytic conductor: \(0.933522\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 14641,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad11 \( 1 \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
3$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.3.c_h
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.23.s_ex
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.31.k_dj
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.37.ao_et
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.59.be_nf
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.67.aba_lr
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.71.g_fv
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.97.abi_sp
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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

−16.77484218834198656532034409063, −16.02369681343710121207109270579, −16.02369681343710121207109270579, −14.85409036669708226539643283017, −14.85409036669708226539643283017, −14.03863540238532335896943207820, −14.03863540238532335896943207820, −12.76419543201372417939560661413, −12.76419543201372417939560661413, −11.90069246142032537610481501027, −11.90069246142032537610481501027, −11.10044716470991944961809074243, −11.10044716470991944961809074243, −9.790262208323322424254691277923, −9.790262208323322424254691277923, −8.487357851440445247270968425966, −8.487357851440445247270968425966, −7.74971489374349908553225834156, −7.74971489374349908553225834156, −6.04214609726225978583223790134, −6.04214609726225978583223790134, −4.73070320716804110076258258958, −4.73070320716804110076258258958, −3.60577326155639270878357097593, −3.60577326155639270878357097593, 0, 0, 3.60577326155639270878357097593, 3.60577326155639270878357097593, 4.73070320716804110076258258958, 4.73070320716804110076258258958, 6.04214609726225978583223790134, 6.04214609726225978583223790134, 7.74971489374349908553225834156, 7.74971489374349908553225834156, 8.487357851440445247270968425966, 8.487357851440445247270968425966, 9.790262208323322424254691277923, 9.790262208323322424254691277923, 11.10044716470991944961809074243, 11.10044716470991944961809074243, 11.90069246142032537610481501027, 11.90069246142032537610481501027, 12.76419543201372417939560661413, 12.76419543201372417939560661413, 14.03863540238532335896943207820, 14.03863540238532335896943207820, 14.85409036669708226539643283017, 14.85409036669708226539643283017, 16.02369681343710121207109270579, 16.02369681343710121207109270579, 16.77484218834198656532034409063

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