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Properties

Label 4-4332e2-1.1-c1e2-0-4
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 − 5-s − 3·7-s + 3·9-s + 10·11-s − 7·13-s + 2·15-s + 6·21-s − 2·23-s − 8·25-s − 4·27-s + 2·29-s − 3·31-s − 20·33-s + 3·35-s + 4·37-s + 14·39-s − 11·41-s − 12·43-s − 3·45-s + 7·47-s + 4·49-s − 6·53-s − 10·55-s + 4·59-s + 22·61-s − 9·63-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 1.13·7-s + 9-s + 3.01·11-s − 1.94·13-s + 0.516·15-s + 1.30·21-s − 0.417·23-s − 8/5·25-s − 0.769·27-s + 0.371·29-s − 0.538·31-s − 3.48·33-s + 0.507·35-s + 0.657·37-s + 2.24·39-s − 1.71·41-s − 1.82·43-s − 0.447·45-s + 1.02·47-s + 4/7·49-s − 0.824·53-s − 1.34·55-s + 0.520·59-s + 2.81·61-s − 1.13·63-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$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_j
7$D_{4}$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_f
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.11.ak_bv
13$D_{4}$ \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.13.h_bl
17$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.17.a_bd
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.23.c_bv
29$D_{4}$ \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_o
31$D_{4}$ \( 1 + 3 T + 33 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.31.d_bh
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$D_{4}$ \( 1 + 11 T + 101 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.41.l_dx
43$D_{4}$ \( 1 + 12 T + 117 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_en
47$D_{4}$ \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.47.ah_bt
53$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_bj
59$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_bq
61$D_{4}$ \( 1 - 22 T + 238 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.61.aw_je
67$D_{4}$ \( 1 + 25 T + 279 T^{2} + 25 p T^{3} + p^{2} T^{4} \) 2.67.z_kt
71$D_{4}$ \( 1 - 17 T + 3 p T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.71.ar_if
73$D_{4}$ \( 1 + 12 T + 177 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.73.m_gv
79$D_{4}$ \( 1 - 4 T + 117 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.79.ae_en
83$D_{4}$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_an
89$D_{4}$ \( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.89.au_jy
97$D_{4}$ \( 1 - 10 T + 214 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_ig
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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

−7.926584267335432294050700089063, −7.912069622306193162538993962861, −7.18702722173009862577202806593, −7.02565772756627155708677218242, −6.58388120030459427408005586895, −6.57003662479641473770226616176, −6.05558770022509648004147460360, −5.76621254304529187989379157894, −5.05742543115746198249776712126, −5.01954784442764497612405997137, −4.32387271824506428673386199609, −4.02861857005196511936148559161, −3.57212964062101671597589518006, −3.56056304914675305225346909773, −2.62307707956609267512539936772, −2.14000463313993038180456815451, −1.51622222916509532555449374980, −1.07583316147788223595744838060, 0, 0, 1.07583316147788223595744838060, 1.51622222916509532555449374980, 2.14000463313993038180456815451, 2.62307707956609267512539936772, 3.56056304914675305225346909773, 3.57212964062101671597589518006, 4.02861857005196511936148559161, 4.32387271824506428673386199609, 5.01954784442764497612405997137, 5.05742543115746198249776712126, 5.76621254304529187989379157894, 6.05558770022509648004147460360, 6.57003662479641473770226616176, 6.58388120030459427408005586895, 7.02565772756627155708677218242, 7.18702722173009862577202806593, 7.912069622306193162538993962861, 7.926584267335432294050700089063

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