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Properties

Label 4-960e2-1.1-c1e2-0-20
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
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  − 4·5-s − 6·7-s − 9-s − 2·11-s + 8·13-s + 6·17-s + 6·19-s + 2·23-s + 11·25-s + 6·29-s + 24·35-s + 16·37-s + 12·43-s + 4·45-s − 10·47-s + 18·49-s + 8·55-s + 14·59-s − 18·61-s + 6·63-s − 32·65-s + 4·67-s + 16·71-s − 10·73-s + 12·77-s + 81-s − 24·85-s + ⋯
L(s)  = 1  − 1.78·5-s − 2.26·7-s − 1/3·9-s − 0.603·11-s + 2.21·13-s + 1.45·17-s + 1.37·19-s + 0.417·23-s + 11/5·25-s + 1.11·29-s + 4.05·35-s + 2.63·37-s + 1.82·43-s + 0.596·45-s − 1.45·47-s + 18/7·49-s + 1.07·55-s + 1.82·59-s − 2.30·61-s + 0.755·63-s − 3.96·65-s + 0.488·67-s + 1.89·71-s − 1.17·73-s + 1.36·77-s + 1/9·81-s − 2.60·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.248585816\)
\(L(\frac12)\) \(\approx\) \(1.248585816\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_s
11$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_c
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ag_s
19$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.19.ag_s
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_c
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.ag_s
31$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.31.a_bm
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_by
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_du
61$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.61.s_gg
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.k_by
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.97.o_du
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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

−10.05666455146787269732823924214, −9.894678045649337563943791647709, −9.409896548029265866693877053678, −8.991645488224489472355898403063, −8.526812315973753380135801107371, −8.080670316971243145538409775595, −7.70936025147659137853429108406, −7.46848789369420650907225442905, −6.74272064537000155325192388997, −6.47966922993428207882949473400, −5.89654118929861915696712341992, −5.74229659407871721222421844457, −4.93322811690000764927892897733, −4.30671189862613375022402353157, −3.72089945689145135046618959406, −3.45924509779825604876518045058, −3.01178230664606519054703495668, −2.77890855266839489798185807409, −0.999524112525943013884931767549, −0.70536687291686079420166310973, 0.70536687291686079420166310973, 0.999524112525943013884931767549, 2.77890855266839489798185807409, 3.01178230664606519054703495668, 3.45924509779825604876518045058, 3.72089945689145135046618959406, 4.30671189862613375022402353157, 4.93322811690000764927892897733, 5.74229659407871721222421844457, 5.89654118929861915696712341992, 6.47966922993428207882949473400, 6.74272064537000155325192388997, 7.46848789369420650907225442905, 7.70936025147659137853429108406, 8.080670316971243145538409775595, 8.526812315973753380135801107371, 8.991645488224489472355898403063, 9.409896548029265866693877053678, 9.894678045649337563943791647709, 10.05666455146787269732823924214

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