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Properties

Label 4-364e2-1.1-c1e2-0-9
Degree $4$
Conductor $132496$
Sign $1$
Analytic cond. $8.44805$
Root an. cond. $1.70486$
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 + 2·7-s + 6·11-s + 2·13-s + 6·17-s + 4·19-s + 4·21-s + 6·23-s − 7·25-s − 2·27-s − 6·29-s − 8·31-s + 12·33-s − 2·37-s + 4·39-s − 14·43-s + 12·47-s + 3·49-s + 12·51-s − 6·53-s + 8·57-s − 20·61-s − 8·67-s + 12·69-s − 6·71-s − 8·73-s − 14·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 1.80·11-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 1.25·23-s − 7/5·25-s − 0.384·27-s − 1.11·29-s − 1.43·31-s + 2.08·33-s − 0.328·37-s + 0.640·39-s − 2.13·43-s + 1.75·47-s + 3/7·49-s + 1.68·51-s − 0.824·53-s + 1.05·57-s − 2.56·61-s − 0.977·67-s + 1.44·69-s − 0.712·71-s − 0.936·73-s − 1.61·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(132496\)    =    \(2^{4} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(8.44805\)
Root analytic conductor: \(1.70486\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 132496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.929934528\)
\(L(\frac12)\) \(\approx\) \(2.929934528\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
17$D_{4}$ \( 1 - 6 T + 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_q
19$D_{4}$ \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_p
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$D_{4}$ \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_cd
31$D_{4}$ \( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bz
37$D_{4}$ \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_bw
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.43.o_ff
47$D_{4}$ \( 1 - 12 T + 127 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_ex
53$D_{4}$ \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_cp
59$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.59.a_k
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_bq
71$D_{4}$ \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_fs
73$D_{4}$ \( 1 + 8 T + 135 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_ff
79$D_{4}$ \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_bz
83$D_{4}$ \( 1 + 12 T + 175 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_gt
89$D_{4}$ \( 1 - 12 T + 187 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_hf
97$D_{4}$ \( 1 - 16 T + 231 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_ix
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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

−11.55354777303285183537828862048, −11.48162942371729126931923445671, −10.62562328018689743867872336853, −10.36084466657607284142720822721, −9.427953620658775691068582805548, −9.343634114620225412128553460420, −8.919002538328781227320795980179, −8.561716837758046088552720669195, −7.80985016415399555533128753352, −7.60300856211437116905583953285, −7.14403758966548007283223252676, −6.39449945158992490277324879979, −5.73070678677033449952453341354, −5.44610844428662567205651774747, −4.58578199167383713369794242349, −3.90423520744475782645598411607, −3.35473243646736378058596595982, −3.08037979132238109847459385699, −1.76859295600357545086886342204, −1.40394088187422053685394686066, 1.40394088187422053685394686066, 1.76859295600357545086886342204, 3.08037979132238109847459385699, 3.35473243646736378058596595982, 3.90423520744475782645598411607, 4.58578199167383713369794242349, 5.44610844428662567205651774747, 5.73070678677033449952453341354, 6.39449945158992490277324879979, 7.14403758966548007283223252676, 7.60300856211437116905583953285, 7.80985016415399555533128753352, 8.561716837758046088552720669195, 8.919002538328781227320795980179, 9.343634114620225412128553460420, 9.427953620658775691068582805548, 10.36084466657607284142720822721, 10.62562328018689743867872336853, 11.48162942371729126931923445671, 11.55354777303285183537828862048

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