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Properties

Label 4-3240e2-1.1-c1e2-0-1
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
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  + 5-s − 4·11-s − 6·13-s + 12·17-s − 8·19-s − 2·29-s + 8·31-s − 4·37-s − 6·41-s − 12·43-s + 8·47-s + 7·49-s − 12·53-s − 4·55-s + 12·59-s − 14·61-s − 6·65-s − 4·67-s − 16·71-s − 12·73-s + 8·79-s − 12·83-s + 12·85-s − 20·89-s − 8·95-s − 2·97-s + 6·101-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s − 1.66·13-s + 2.91·17-s − 1.83·19-s − 0.371·29-s + 1.43·31-s − 0.657·37-s − 0.937·41-s − 1.82·43-s + 1.16·47-s + 49-s − 1.64·53-s − 0.539·55-s + 1.56·59-s − 1.79·61-s − 0.744·65-s − 0.488·67-s − 1.89·71-s − 1.40·73-s + 0.900·79-s − 1.31·83-s + 1.30·85-s − 2.11·89-s − 0.820·95-s − 0.203·97-s + 0.597·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6468931247\)
\(L(\frac12)\) \(\approx\) \(0.6468931247\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_f
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_az
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_bh
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2^2$ \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dx
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.47.ai_r
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_dh
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.o_ff
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_ap
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−8.910209269918978888702986569809, −8.322873086338119079293919003831, −8.045604733230375329004763712853, −7.76757461116773140042942922589, −7.48022522321709425468438581927, −6.85093745763670635652888586003, −6.81364482004581152325026080271, −6.06288676651208876461685178181, −5.77736851132505997741793556810, −5.32497155985260880778888837747, −5.22566176331012922434777711248, −4.53925202465065590897985155129, −4.40213612816041739260271274480, −3.64814705817298084049408065344, −3.16836673561107486553005468021, −2.62052270679392534567657702583, −2.60915093873376136705339227485, −1.65092101341958065858140906244, −1.39097051699525463643328629306, −0.23843500139855372075214458488, 0.23843500139855372075214458488, 1.39097051699525463643328629306, 1.65092101341958065858140906244, 2.60915093873376136705339227485, 2.62052270679392534567657702583, 3.16836673561107486553005468021, 3.64814705817298084049408065344, 4.40213612816041739260271274480, 4.53925202465065590897985155129, 5.22566176331012922434777711248, 5.32497155985260880778888837747, 5.77736851132505997741793556810, 6.06288676651208876461685178181, 6.81364482004581152325026080271, 6.85093745763670635652888586003, 7.48022522321709425468438581927, 7.76757461116773140042942922589, 8.045604733230375329004763712853, 8.322873086338119079293919003831, 8.910209269918978888702986569809

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