Properties

Label 4-429e2-1.1-c1e2-0-6
Degree $4$
Conductor $184041$
Sign $1$
Analytic cond. $11.7346$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·4-s + 4·5-s − 7-s + 11-s + 2·12-s + 7·13-s + 4·15-s − 2·17-s + 4·19-s + 8·20-s − 21-s − 4·23-s + 2·25-s − 27-s − 2·28-s + 8·29-s − 6·31-s + 33-s − 4·35-s − 10·37-s + 7·39-s + 8·41-s + 7·43-s + 2·44-s + 16·47-s + 7·49-s + ⋯
L(s)  = 1  + 0.577·3-s + 4-s + 1.78·5-s − 0.377·7-s + 0.301·11-s + 0.577·12-s + 1.94·13-s + 1.03·15-s − 0.485·17-s + 0.917·19-s + 1.78·20-s − 0.218·21-s − 0.834·23-s + 2/5·25-s − 0.192·27-s − 0.377·28-s + 1.48·29-s − 1.07·31-s + 0.174·33-s − 0.676·35-s − 1.64·37-s + 1.12·39-s + 1.24·41-s + 1.06·43-s + 0.301·44-s + 2.33·47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(184041\)    =    \(3^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(11.7346\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 184041,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.735496644\)
\(L(\frac12)\) \(\approx\) \(3.735496644\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_2$ \( 1 - T + T^{2} \)
11$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 - 7 T + p T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_an
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_ad
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_ah
29$C_2^2$ \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.29.ai_bj
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.k_cl
41$C_2^2$ \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_x
43$C_2^2$ \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.43.ah_g
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2^2$ \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_f
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.n_ee
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.l_cc
71$C_2^2$ \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_abj
73$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.73.ba_md
79$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.79.o_hz
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2^2$ \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.89.i_az
97$C_2^2$ \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.97.ah_abw
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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

−11.30595891049469317905434889227, −10.75274073764671175040568899616, −10.45307335529825739568476806661, −10.24500098285570941431861656169, −9.276967248628663996638134695676, −9.235695679783324700987368205497, −8.949181378392791099305342860737, −8.282620266404962473121823690718, −7.53221004248768402197641140024, −7.30985308756402453078586037727, −6.47544618750378953581654077498, −6.21825399260149417762678334892, −5.77268977362549371154352940587, −5.62377635140552850755813229552, −4.40551627593841665537675363006, −3.96552228884396473062134756680, −2.93829981892094922936304038961, −2.81608133806649208677393965654, −1.64427360496605784138222353014, −1.63337268995724863201475421823, 1.63337268995724863201475421823, 1.64427360496605784138222353014, 2.81608133806649208677393965654, 2.93829981892094922936304038961, 3.96552228884396473062134756680, 4.40551627593841665537675363006, 5.62377635140552850755813229552, 5.77268977362549371154352940587, 6.21825399260149417762678334892, 6.47544618750378953581654077498, 7.30985308756402453078586037727, 7.53221004248768402197641140024, 8.282620266404962473121823690718, 8.949181378392791099305342860737, 9.235695679783324700987368205497, 9.276967248628663996638134695676, 10.24500098285570941431861656169, 10.45307335529825739568476806661, 10.75274073764671175040568899616, 11.30595891049469317905434889227

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