Properties

Label 4-11e4-1.1-c1e2-0-0
Degree $4$
Conductor $14641$
Sign $1$
Analytic cond. $0.933522$
Root an. cond. $0.982949$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 3·4-s + 2·5-s + 6·9-s − 12·12-s + 8·15-s + 5·16-s − 6·20-s + 4·23-s − 7·25-s − 4·27-s − 4·31-s − 18·36-s − 6·37-s + 12·45-s + 4·47-s + 20·48-s − 10·49-s + 18·53-s + 16·59-s − 24·60-s − 3·64-s + 4·67-s + 16·69-s + 24·71-s − 28·75-s + 10·80-s + ⋯
L(s)  = 1  + 2.30·3-s − 3/2·4-s + 0.894·5-s + 2·9-s − 3.46·12-s + 2.06·15-s + 5/4·16-s − 1.34·20-s + 0.834·23-s − 7/5·25-s − 0.769·27-s − 0.718·31-s − 3·36-s − 0.986·37-s + 1.78·45-s + 0.583·47-s + 2.88·48-s − 1.42·49-s + 2.47·53-s + 2.08·59-s − 3.09·60-s − 3/8·64-s + 0.488·67-s + 1.92·69-s + 2.84·71-s − 3.23·75-s + 1.11·80-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: \(0\)
Selberg data: \((4,\ 14641,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.699138266\)
\(L(\frac12)\) \(\approx\) \(1.699138266\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.17.a_j
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.23.ae_by
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.41.a_cf
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.53.as_hf
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.59.aq_ha
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.a_di
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.79.a_cg
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.97.ba_nz
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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

−11.10272305542928838383025084052, −9.975450561586109317601560382100, −9.950218896470013175355990146225, −9.182457967238804819066801847611, −9.127942331541421284163370204249, −8.356936232923193217562617522792, −8.264856530445176716222542589828, −7.49323810887805876503518242868, −6.72221402891887535126565651102, −5.44861874763839175210229504778, −5.38415572352897256696127903868, −3.89751875979123562820900427499, −3.84177902362047353880486406829, −2.75274084463573300677827364197, −1.98218366620434510274683932709, 1.98218366620434510274683932709, 2.75274084463573300677827364197, 3.84177902362047353880486406829, 3.89751875979123562820900427499, 5.38415572352897256696127903868, 5.44861874763839175210229504778, 6.72221402891887535126565651102, 7.49323810887805876503518242868, 8.264856530445176716222542589828, 8.356936232923193217562617522792, 9.127942331541421284163370204249, 9.182457967238804819066801847611, 9.950218896470013175355990146225, 9.975450561586109317601560382100, 11.10272305542928838383025084052

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