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Properties

Label 4-4080e2-1.1-c1e2-0-25
Degree $4$
Conductor $16646400$
Sign $1$
Analytic cond. $1061.38$
Root an. cond. $5.70779$
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  + 2·3-s + 2·5-s − 7-s + 3·9-s + 7·11-s − 2·13-s + 4·15-s + 2·17-s − 7·19-s − 2·21-s + 3·25-s + 4·27-s + 11·29-s + 14·33-s − 2·35-s + 11·37-s − 4·39-s + 3·41-s − 8·43-s + 6·45-s + 17·47-s − 5·49-s + 4·51-s + 9·53-s + 14·55-s − 14·57-s − 6·59-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 0.377·7-s + 9-s + 2.11·11-s − 0.554·13-s + 1.03·15-s + 0.485·17-s − 1.60·19-s − 0.436·21-s + 3/5·25-s + 0.769·27-s + 2.04·29-s + 2.43·33-s − 0.338·35-s + 1.80·37-s − 0.640·39-s + 0.468·41-s − 1.21·43-s + 0.894·45-s + 2.47·47-s − 5/7·49-s + 0.560·51-s + 1.23·53-s + 1.88·55-s − 1.85·57-s − 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16646400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16646400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1061.38\)
Root analytic conductor: \(5.70779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16646400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.651439208\)
\(L(\frac12)\) \(\approx\) \(7.651439208\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_g
11$D_{4}$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.11.ah_ba
13$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_ag
19$D_{4}$ \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.19.h_bq
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$D_{4}$ \( 1 - 11 T + 80 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.29.al_dc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$D_{4}$ \( 1 - 11 T + 96 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.37.al_ds
41$D_{4}$ \( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_cy
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$D_{4}$ \( 1 - 17 T + 158 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.47.ar_gc
53$D_{4}$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_ca
59$D_{4}$ \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.59.g_dq
61$D_{4}$ \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_ek
67$D_{4}$ \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_eg
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$D_{4}$ \( 1 - 5 T + 144 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.73.af_fo
79$D_{4}$ \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_ew
83$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_bm
89$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.89.i_ck
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−8.629294784759904751423681599103, −8.465539062175428705303054869182, −7.80080084117607326486211796799, −7.74205612318810657670186338013, −6.89374863860195173218563781468, −6.85093448846986111948053621903, −6.45442722009474455830055561969, −6.27267211867829412130172921043, −5.68934042195116440685465664256, −5.36313727671195023067343601645, −4.51798696291463088924994511595, −4.46102048702170279135878254534, −4.07988887387666294104250695958, −3.60624628769672871015199610679, −3.08923817997205572921977951897, −2.70585435802875810372614594035, −2.06701810120027231198504403006, −2.05688727313117556416488292186, −1.04939055655392916419770537390, −0.882719482990355179571335541345, 0.882719482990355179571335541345, 1.04939055655392916419770537390, 2.05688727313117556416488292186, 2.06701810120027231198504403006, 2.70585435802875810372614594035, 3.08923817997205572921977951897, 3.60624628769672871015199610679, 4.07988887387666294104250695958, 4.46102048702170279135878254534, 4.51798696291463088924994511595, 5.36313727671195023067343601645, 5.68934042195116440685465664256, 6.27267211867829412130172921043, 6.45442722009474455830055561969, 6.85093448846986111948053621903, 6.89374863860195173218563781468, 7.74205612318810657670186338013, 7.80080084117607326486211796799, 8.465539062175428705303054869182, 8.629294784759904751423681599103

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