Properties

Label 4-9702e2-1.1-c1e2-0-21
Degree $4$
Conductor $94128804$
Sign $1$
Analytic cond. $6001.73$
Root an. cond. $8.80175$
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  − 2·2-s + 3·4-s − 4·8-s + 2·11-s − 4·13-s + 5·16-s + 8·17-s − 4·19-s − 4·22-s + 8·26-s − 8·29-s − 12·31-s − 6·32-s − 16·34-s + 4·37-s + 8·38-s − 8·41-s + 6·44-s + 4·47-s − 12·52-s + 4·53-s + 16·58-s − 16·59-s − 20·61-s + 24·62-s + 7·64-s − 4·67-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 0.603·11-s − 1.10·13-s + 5/4·16-s + 1.94·17-s − 0.917·19-s − 0.852·22-s + 1.56·26-s − 1.48·29-s − 2.15·31-s − 1.06·32-s − 2.74·34-s + 0.657·37-s + 1.29·38-s − 1.24·41-s + 0.904·44-s + 0.583·47-s − 1.66·52-s + 0.549·53-s + 2.10·58-s − 2.08·59-s − 2.56·61-s + 3.04·62-s + 7/8·64-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(94128804\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(6001.73\)
Root analytic conductor: \(8.80175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 94128804,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + p^{2} T^{4} \) 2.5.a_a
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bo
19$D_{4}$ \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_bg
23$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.23.a_g
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
31$D_{4}$ \( 1 + 12 T + 88 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_dk
37$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_bm
41$D_{4}$ \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_dk
43$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.43.a_bu
47$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_i
53$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_cs
59$D_{4}$ \( 1 + 16 T + 142 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_fm
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$D_{4}$ \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_du
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$D_{4}$ \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_cu
79$D_{4}$ \( 1 - 4 T + 122 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_es
83$D_{4}$ \( 1 + 4 T + 160 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_ge
89$D_{4}$ \( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.89.au_je
97$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_bm
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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

−7.43229212867977654631863829041, −7.38531929091118747658082890629, −7.05100878520088292215705119986, −6.64367639241421938404189257013, −6.09039797233091423340140884663, −5.98594681948127232610011831253, −5.56078678109775786021304887012, −5.28774037765591845899728077674, −4.69280642360611892020918133215, −4.48688768950549145088372343491, −3.67614353184383221862693969832, −3.62820098846160066491777629989, −3.17366923453831242889919785392, −2.74595065166406688370617065781, −2.04329271972541695047603480238, −1.96011583243910597231882511717, −1.41167510424924385001235933117, −0.972660771939041836566623316108, 0, 0, 0.972660771939041836566623316108, 1.41167510424924385001235933117, 1.96011583243910597231882511717, 2.04329271972541695047603480238, 2.74595065166406688370617065781, 3.17366923453831242889919785392, 3.62820098846160066491777629989, 3.67614353184383221862693969832, 4.48688768950549145088372343491, 4.69280642360611892020918133215, 5.28774037765591845899728077674, 5.56078678109775786021304887012, 5.98594681948127232610011831253, 6.09039797233091423340140884663, 6.64367639241421938404189257013, 7.05100878520088292215705119986, 7.38531929091118747658082890629, 7.43229212867977654631863829041

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