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Properties

Label 4-2790e2-1.1-c1e2-0-11
Degree $4$
Conductor $7784100$
Sign $1$
Analytic cond. $496.320$
Root an. cond. $4.71998$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 2·5-s + 7-s − 4·8-s + 4·10-s − 11-s + 4·13-s − 2·14-s + 5·16-s − 2·17-s − 3·19-s − 6·20-s + 2·22-s − 7·23-s + 3·25-s − 8·26-s + 3·28-s − 6·29-s + 2·31-s − 6·32-s + 4·34-s − 2·35-s + 2·37-s + 6·38-s + 8·40-s − 6·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.377·7-s − 1.41·8-s + 1.26·10-s − 0.301·11-s + 1.10·13-s − 0.534·14-s + 5/4·16-s − 0.485·17-s − 0.688·19-s − 1.34·20-s + 0.426·22-s − 1.45·23-s + 3/5·25-s − 1.56·26-s + 0.566·28-s − 1.11·29-s + 0.359·31-s − 1.06·32-s + 0.685·34-s − 0.338·35-s + 0.328·37-s + 0.973·38-s + 1.26·40-s − 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7784100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7784100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(496.320\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 7784100,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_k
11$D_{4}$ \( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_s
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_s
19$D_{4}$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_c
23$D_{4}$ \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.23.h_cc
29$D_{4}$ \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_by
37$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_cg
41$D_{4}$ \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_cw
43$D_{4}$ \( 1 - 5 T - 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.43.af_ao
47$D_{4}$ \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.47.c_da
53$D_{4}$ \( 1 + 19 T + 192 T^{2} + 19 p T^{3} + p^{2} T^{4} \) 2.53.t_hk
59$D_{4}$ \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.59.as_ha
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$D_{4}$ \( 1 + 6 T - 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_ak
71$D_{4}$ \( 1 - 3 T + 106 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.71.ad_ec
73$D_{4}$ \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) 2.73.b_bo
79$D_{4}$ \( 1 + 21 T + 230 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.79.v_iw
83$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_fe
89$D_{4}$ \( 1 - 15 T + 196 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_ho
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−8.380274100276154634723532407800, −8.354743832392681783151520635946, −7.889559608806654526613142391429, −7.84840495452785283897936671622, −7.18510630725992642247790849893, −6.80986850271746098711925464653, −6.47159808835076253954481404234, −6.18348433413767279723290568307, −5.47505764581384809578360766360, −5.40386405569995383148600131527, −4.45665963167379737437363568466, −4.31215541623833777994529899236, −3.56624755388065188473922082707, −3.53523513964495048951777514468, −2.55556984015940836652716599214, −2.37587517893091441460462867653, −1.46553348244527503199022869130, −1.33302563099350497200222708063, 0, 0, 1.33302563099350497200222708063, 1.46553348244527503199022869130, 2.37587517893091441460462867653, 2.55556984015940836652716599214, 3.53523513964495048951777514468, 3.56624755388065188473922082707, 4.31215541623833777994529899236, 4.45665963167379737437363568466, 5.40386405569995383148600131527, 5.47505764581384809578360766360, 6.18348433413767279723290568307, 6.47159808835076253954481404234, 6.80986850271746098711925464653, 7.18510630725992642247790849893, 7.84840495452785283897936671622, 7.889559608806654526613142391429, 8.354743832392681783151520635946, 8.380274100276154634723532407800

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