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Properties

Label 4-1452e2-1.1-c1e2-0-7
Degree $4$
Conductor $2108304$
Sign $1$
Analytic cond. $134.427$
Root an. cond. $3.40503$
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  − 2·3-s − 5-s − 4·7-s + 3·9-s + 8·13-s + 2·15-s − 7·17-s − 19-s + 8·21-s + 4·23-s + 2·25-s − 4·27-s − 15·31-s + 4·35-s − 12·37-s − 16·39-s − 10·41-s − 10·43-s − 3·45-s + 5·47-s + 3·49-s + 14·51-s − 7·53-s + 2·57-s + 11·59-s + 3·61-s − 12·63-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 1.51·7-s + 9-s + 2.21·13-s + 0.516·15-s − 1.69·17-s − 0.229·19-s + 1.74·21-s + 0.834·23-s + 2/5·25-s − 0.769·27-s − 2.69·31-s + 0.676·35-s − 1.97·37-s − 2.56·39-s − 1.56·41-s − 1.52·43-s − 0.447·45-s + 0.729·47-s + 3/7·49-s + 1.96·51-s − 0.961·53-s + 0.264·57-s + 1.43·59-s + 0.384·61-s − 1.51·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2108304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2108304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2108304\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(134.427\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2108304,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good5$D_{4}$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ab
7$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_n
13$D_{4}$ \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bl
17$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bt
19$D_{4}$ \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) 2.19.b_h
23$D_{4}$ \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bt
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.29.a_bm
31$D_{4}$ \( 1 + 15 T + 117 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.31.p_en
37$D_{4}$ \( 1 + 12 T + 105 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_eb
41$D_{4}$ \( 1 + 10 T + 87 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_dj
43$D_{4}$ \( 1 + 10 T + 91 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_dn
47$D_{4}$ \( 1 - 5 T + 99 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.47.af_dv
53$D_{4}$ \( 1 + 7 T + 17 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.53.h_r
59$D_{4}$ \( 1 - 11 T + 137 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.59.al_fh
61$D_{4}$ \( 1 - 3 T + 123 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.61.ad_et
67$D_{4}$ \( 1 + 7 T + 135 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_ff
71$D_{4}$ \( 1 + 5 T + 147 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.71.f_fr
73$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.73.c_fm
79$D_{4}$ \( 1 + 24 T + 297 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.79.y_ll
83$D_{4}$ \( 1 + 22 T + 267 T^{2} + 22 p T^{3} + p^{2} T^{4} \) 2.83.w_kh
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.89.ac_gx
97$D_{4}$ \( 1 - T + 163 T^{2} - p T^{3} + p^{2} T^{4} \) 2.97.ab_gh
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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

−9.154330006460896463712885755248, −8.830341188319842416553234282663, −8.696544972878169955011376847445, −8.380270746556061855414968287281, −7.40973026357062656699246483417, −7.07564354999511872108101785371, −6.86299248002417109606322716945, −6.49144618170475771076018359584, −6.03947742551539057878635265695, −5.76966136436514785605235865344, −5.12125669641262378405058505834, −4.86290527847168790090088235763, −3.96935515445952591227812487474, −3.83075727509495184414864371535, −3.38439668111259848109782356332, −2.84533115040378659662064183963, −1.74961616055768535065621824826, −1.41224756078092261457138151205, 0, 0, 1.41224756078092261457138151205, 1.74961616055768535065621824826, 2.84533115040378659662064183963, 3.38439668111259848109782356332, 3.83075727509495184414864371535, 3.96935515445952591227812487474, 4.86290527847168790090088235763, 5.12125669641262378405058505834, 5.76966136436514785605235865344, 6.03947742551539057878635265695, 6.49144618170475771076018359584, 6.86299248002417109606322716945, 7.07564354999511872108101785371, 7.40973026357062656699246483417, 8.380270746556061855414968287281, 8.696544972878169955011376847445, 8.830341188319842416553234282663, 9.154330006460896463712885755248

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