Properties

Label 4-2738e2-1.1-c1e2-0-3
Degree $4$
Conductor $7496644$
Sign $1$
Analytic cond. $477.992$
Root an. cond. $4.67579$
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-s + 3·4-s − 5-s + 2·6-s − 2·7-s − 4·8-s − 4·9-s + 2·10-s − 5·11-s − 3·12-s − 13-s + 4·14-s + 15-s + 5·16-s + 8·18-s − 3·20-s + 2·21-s + 10·22-s + 23-s + 4·24-s + 2·25-s + 2·26-s + 6·27-s − 6·28-s + 3·29-s − 2·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.41·8-s − 4/3·9-s + 0.632·10-s − 1.50·11-s − 0.866·12-s − 0.277·13-s + 1.06·14-s + 0.258·15-s + 5/4·16-s + 1.88·18-s − 0.670·20-s + 0.436·21-s + 2.13·22-s + 0.208·23-s + 0.816·24-s + 2/5·25-s + 0.392·26-s + 1.15·27-s − 1.13·28-s + 0.557·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7496644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7496644\)    =    \(2^{2} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(477.992\)
Root analytic conductor: \(4.67579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7496644,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
37 \( 1 \)
good3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_f
5$D_{4}$ \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ab
7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_k
11$D_{4}$ \( 1 + 5 T + 27 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_bb
13$D_{4}$ \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_p
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$D_{4}$ \( 1 - T + 35 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_bj
29$C_4$ \( 1 - 3 T - T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_ab
31$D_{4}$ \( 1 + 17 T + 133 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.31.r_fd
41$D_{4}$ \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.41.ar_fx
43$D_{4}$ \( 1 - 6 T + 90 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_dm
47$D_{4}$ \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.47.ac_dm
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_dy
59$D_{4}$ \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_gg
61$D_{4}$ \( 1 + 19 T + 211 T^{2} + 19 p T^{3} + p^{2} T^{4} \) 2.61.t_id
67$D_{4}$ \( 1 + 9 T + 123 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.67.j_et
71$D_{4}$ \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_du
73$D_{4}$ \( 1 - 3 T + 117 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.73.ad_en
79$D_{4}$ \( 1 + 3 T + 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.79.d_ch
83$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.83.u_jm
89$C_4$ \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_hm
97$D_{4}$ \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hi
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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

−8.685255636484420695268696989944, −8.324865818936134346704409798511, −7.80964047203371484422591504252, −7.53477679154028796213509666879, −7.35504701531525281159399632747, −6.92546084225036100619064195053, −6.28621039910627201463811703134, −6.00224362486984751971898444430, −5.58134954053315061259743451501, −5.51572598613675427335068081973, −4.74943885482265123249048046325, −4.37511142431396912285236867155, −3.56372481801532921799683077427, −3.14076060560635594142021805618, −2.78160992546849929633713767929, −2.41924354728625773119860485800, −1.73712553269362142523933281378, −0.841040212647889374715012444808, 0, 0, 0.841040212647889374715012444808, 1.73712553269362142523933281378, 2.41924354728625773119860485800, 2.78160992546849929633713767929, 3.14076060560635594142021805618, 3.56372481801532921799683077427, 4.37511142431396912285236867155, 4.74943885482265123249048046325, 5.51572598613675427335068081973, 5.58134954053315061259743451501, 6.00224362486984751971898444430, 6.28621039910627201463811703134, 6.92546084225036100619064195053, 7.35504701531525281159399632747, 7.53477679154028796213509666879, 7.80964047203371484422591504252, 8.324865818936134346704409798511, 8.685255636484420695268696989944

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