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Properties

Label 4-1344e2-1.1-c1e2-0-39
Degree $4$
Conductor $1806336$
Sign $1$
Analytic cond. $115.173$
Root an. cond. $3.27595$
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 + 5-s + 7-s + 5·11-s − 15-s + 4·17-s + 8·19-s − 21-s + 4·23-s + 5·25-s + 27-s + 10·29-s − 3·31-s − 5·33-s + 35-s − 4·37-s − 4·43-s + 6·47-s − 6·49-s − 4·51-s − 9·53-s + 5·55-s − 8·57-s − 11·59-s − 6·61-s − 2·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1.50·11-s − 0.258·15-s + 0.970·17-s + 1.83·19-s − 0.218·21-s + 0.834·23-s + 25-s + 0.192·27-s + 1.85·29-s − 0.538·31-s − 0.870·33-s + 0.169·35-s − 0.657·37-s − 0.609·43-s + 0.875·47-s − 6/7·49-s − 0.560·51-s − 1.23·53-s + 0.674·55-s − 1.05·57-s − 1.43·59-s − 0.768·61-s − 0.244·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(115.173\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.923887369\)
\(L(\frac12)\) \(\approx\) \(2.923887369\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_o
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_ab
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.ai_bt
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.29.ak_df
31$C_2^2$ \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.31.d_aw
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_av
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.43.e_dm
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_al
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_bc
59$C_2^2$ \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.59.l_ck
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_az
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_acl
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.k_bb
79$C_2^2$ \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.79.d_acs
83$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.83.ao_ih
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_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

−9.703020070677481457449985748367, −9.498223138964229223370296265003, −9.042556911183410788568345235482, −8.669353540586289696661026013304, −8.316785648138920280600102204253, −7.63989544819864890239188923328, −7.31814192506636318218101810727, −7.00570008998866287603378591154, −6.39970955248528460265734472871, −6.10967083808758111012858445136, −5.77572975485209945145753262008, −5.04818452485235001522858362991, −4.82175612564775978339069600072, −4.57353053222339915448115060498, −3.51829934973211870491450887760, −3.33703807115261084534040867395, −2.87041644261094250076542035864, −1.86381301376214043548103418804, −1.25290607224779742412771481131, −0.887446126835043527403270575874, 0.887446126835043527403270575874, 1.25290607224779742412771481131, 1.86381301376214043548103418804, 2.87041644261094250076542035864, 3.33703807115261084534040867395, 3.51829934973211870491450887760, 4.57353053222339915448115060498, 4.82175612564775978339069600072, 5.04818452485235001522858362991, 5.77572975485209945145753262008, 6.10967083808758111012858445136, 6.39970955248528460265734472871, 7.00570008998866287603378591154, 7.31814192506636318218101810727, 7.63989544819864890239188923328, 8.316785648138920280600102204253, 8.669353540586289696661026013304, 9.042556911183410788568345235482, 9.498223138964229223370296265003, 9.703020070677481457449985748367

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