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Properties

Label 4-6480e2-1.1-c1e2-0-10
Degree $4$
Conductor $41990400$
Sign $1$
Analytic cond. $2677.34$
Root an. cond. $7.19326$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s + 5·11-s + 5·13-s − 4·17-s − 2·19-s − 7·23-s + 3·25-s − 5·29-s − 5·31-s + 2·35-s − 6·37-s − 8·43-s + 5·47-s + 49-s − 9·53-s − 10·55-s − 26·59-s + 2·61-s − 10·65-s + 14·67-s − 9·71-s − 5·77-s − 6·79-s + 10·83-s + 8·85-s − 3·89-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 1.50·11-s + 1.38·13-s − 0.970·17-s − 0.458·19-s − 1.45·23-s + 3/5·25-s − 0.928·29-s − 0.898·31-s + 0.338·35-s − 0.986·37-s − 1.21·43-s + 0.729·47-s + 1/7·49-s − 1.23·53-s − 1.34·55-s − 3.38·59-s + 0.256·61-s − 1.24·65-s + 1.71·67-s − 1.06·71-s − 0.569·77-s − 0.675·79-s + 1.09·83-s + 0.867·85-s − 0.317·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(41990400\)    =    \(2^{8} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2677.34\)
Root analytic conductor: \(7.19326\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 41990400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 + T + p T^{3} + p^{2} T^{4} \) 2.7.b_a
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.11.af_o
13$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.13.af_s
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$D_{4}$ \( 1 + 7 T + 44 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.23.h_bs
29$D_{4}$ \( 1 + 5 T + 50 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.29.f_by
31$D_{4}$ \( 1 + 5 T + 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.31.f_cc
37$D_{4}$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.37.g_ba
41$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.41.a_z
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$D_{4}$ \( 1 - 5 T + 86 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.47.af_di
53$D_{4}$ \( 1 + 9 T + 112 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_ei
59$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.59.ba_lb
61$D_{4}$ \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.61.ac_co
67$D_{4}$ \( 1 - 14 T + 126 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_ew
71$D_{4}$ \( 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.71.j_fs
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.73.a_ade
79$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_eg
83$D_{4}$ \( 1 - 10 T + 134 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_fe
89$D_{4}$ \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.89.d_ca
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.97.abg_ri
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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

−7.72531370044496940801523440608, −7.63648193930740373867045325444, −7.02622982415089490577114519858, −6.78539329163521561807921330971, −6.34459203935677027531652839907, −6.27881928045225106749717240485, −5.80956872165828048441611625271, −5.46205575217073485525597760758, −4.69385241076572364459532301059, −4.64395894262850131818030697995, −3.99134488736222242159192807722, −3.86293974064445782655480772004, −3.47446982358748978578412285007, −3.27710965840970815592660045722, −2.51501938728600943585675844108, −1.91306684777512295504685932627, −1.59356776087878332385941049702, −1.12539533610763178704608994954, 0, 0, 1.12539533610763178704608994954, 1.59356776087878332385941049702, 1.91306684777512295504685932627, 2.51501938728600943585675844108, 3.27710965840970815592660045722, 3.47446982358748978578412285007, 3.86293974064445782655480772004, 3.99134488736222242159192807722, 4.64395894262850131818030697995, 4.69385241076572364459532301059, 5.46205575217073485525597760758, 5.80956872165828048441611625271, 6.27881928045225106749717240485, 6.34459203935677027531652839907, 6.78539329163521561807921330971, 7.02622982415089490577114519858, 7.63648193930740373867045325444, 7.72531370044496940801523440608

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