↑

Properties

Label 4-5440e2-1.1-c1e2-0-7
Degree $4$
Conductor $29593600$
Sign $1$
Analytic cond. $1886.91$
Root an. cond. $6.59079$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s − 2·7-s − 6·11-s + 8·13-s + 4·15-s − 2·17-s − 4·19-s + 4·21-s − 6·23-s + 3·25-s + 2·27-s + 10·31-s + 12·33-s + 4·35-s + 8·37-s − 16·39-s + 8·43-s + 12·47-s − 8·49-s + 4·51-s − 12·53-s + 12·55-s + 8·57-s − 12·59-s − 4·61-s − 16·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.755·7-s − 1.80·11-s + 2.21·13-s + 1.03·15-s − 0.485·17-s − 0.917·19-s + 0.872·21-s − 1.25·23-s + 3/5·25-s + 0.384·27-s + 1.79·31-s + 2.08·33-s + 0.676·35-s + 1.31·37-s − 2.56·39-s + 1.21·43-s + 1.75·47-s − 8/7·49-s + 0.560·51-s − 1.64·53-s + 1.61·55-s + 1.05·57-s − 1.56·59-s − 0.512·61-s − 1.98·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29593600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29593600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29593600\)    =    \(2^{12} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1886.91\)
Root analytic conductor: \(6.59079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 29593600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_m
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_bc
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
19$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_be
23$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_bc
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.29.a_bu
31$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_dg
37$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_da
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_dm
47$D_{4}$ \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_de
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$D_{4}$ \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fm
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_da
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.67.au_ja
71$D_{4}$ \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_cy
73$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_cc
79$D_{4}$ \( 1 + 2 T - 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_adg
83$D_{4}$ \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.83.y_lm
89$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_ec
97$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_fu
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.959314497829689262731123948913, −7.80970495556761182969265867875, −7.26873635428272206285636450313, −6.78969780910536162205506448834, −6.34207917269186727188265448310, −6.17051922441214432915579386873, −5.89855950290715254509175412214, −5.70482360446289719363957453459, −5.08523908256143074815478226172, −4.67783571251073866122118633933, −4.19445357086518781634603704902, −4.15574795140059025801192735322, −3.43302391280414622716487191873, −3.15432788525439262464799657290, −2.54079020490174198820598195275, −2.35303250384610630072168019870, −1.37736621635046325012614634568, −0.866237596936142564823046436795, 0, 0, 0.866237596936142564823046436795, 1.37736621635046325012614634568, 2.35303250384610630072168019870, 2.54079020490174198820598195275, 3.15432788525439262464799657290, 3.43302391280414622716487191873, 4.15574795140059025801192735322, 4.19445357086518781634603704902, 4.67783571251073866122118633933, 5.08523908256143074815478226172, 5.70482360446289719363957453459, 5.89855950290715254509175412214, 6.17051922441214432915579386873, 6.34207917269186727188265448310, 6.78969780910536162205506448834, 7.26873635428272206285636450313, 7.80970495556761182969265867875, 7.959314497829689262731123948913

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