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Properties

Label 4-4332e2-1.1-c1e2-0-7
Degree $4$
Conductor $18766224$
Sign $1$
Analytic cond. $1196.55$
Root an. cond. $5.88142$
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  − 2·3-s + 3·5-s + 7-s + 3·9-s − 3·11-s − 4·13-s − 6·15-s − 3·17-s − 2·21-s − 6·23-s + 5·25-s − 4·27-s + 6·29-s + 2·31-s + 6·33-s + 3·35-s + 2·37-s + 8·39-s + 43-s + 9·45-s − 21·47-s − 5·49-s + 6·51-s − 6·53-s − 9·55-s − 11·61-s + 3·63-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.34·5-s + 0.377·7-s + 9-s − 0.904·11-s − 1.10·13-s − 1.54·15-s − 0.727·17-s − 0.436·21-s − 1.25·23-s + 25-s − 0.769·27-s + 1.11·29-s + 0.359·31-s + 1.04·33-s + 0.507·35-s + 0.328·37-s + 1.28·39-s + 0.152·43-s + 1.34·45-s − 3.06·47-s − 5/7·49-s + 0.840·51-s − 0.824·53-s − 1.21·55-s − 1.40·61-s + 0.377·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18766224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18766224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18766224\)    =    \(2^{4} \cdot 3^{2} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1196.55\)
Root analytic conductor: \(5.88142\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 18766224,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
19 \( 1 \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_e
7$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_g
11$D_{4}$ \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_q
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$D_{4}$ \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_bc
23$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_w
29$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_bi
31$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.31.ac_be
37$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_bq
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$D_{4}$ \( 1 - T + 78 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_da
47$D_{4}$ \( 1 + 21 T + 196 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.47.v_ho
53$D_{4}$ \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_de
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$D_{4}$ \( 1 + 11 T + 144 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.61.l_fo
67$D_{4}$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_g
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$D_{4}$ \( 1 + 5 T + 144 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.73.f_fo
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$D_{4}$ \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_fm
89$D_{4}$ \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.89.s_is
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−8.050646541593140090993045003027, −8.011814690192581646452999176922, −7.39979168675635253853202772206, −6.92218591005918888564761396205, −6.54603966746615509061132086353, −6.47611431909828961903283960411, −5.86707677960731794791175033387, −5.73314161461945930709778273001, −5.11416435410115028082791672044, −4.97926718386849109759038590926, −4.62410838392360118694727898643, −4.31141594372513825832256987184, −3.62568028186427221679741725420, −2.96717633585607887382958404266, −2.52967651786622776664757083508, −2.24399097707562370008520601385, −1.47019775057367440361027638531, −1.37400670346746014911813620070, 0, 0, 1.37400670346746014911813620070, 1.47019775057367440361027638531, 2.24399097707562370008520601385, 2.52967651786622776664757083508, 2.96717633585607887382958404266, 3.62568028186427221679741725420, 4.31141594372513825832256987184, 4.62410838392360118694727898643, 4.97926718386849109759038590926, 5.11416435410115028082791672044, 5.73314161461945930709778273001, 5.86707677960731794791175033387, 6.47611431909828961903283960411, 6.54603966746615509061132086353, 6.92218591005918888564761396205, 7.39979168675635253853202772206, 8.011814690192581646452999176922, 8.050646541593140090993045003027

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