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Properties

Label 4-72e3-1.1-c1e2-0-21
Degree $4$
Conductor $373248$
Sign $-1$
Analytic cond. $23.7986$
Root an. cond. $2.20870$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 2·11-s − 2·17-s + 4·19-s − 3·25-s − 27-s − 2·33-s − 6·41-s − 10·43-s − 5·49-s + 2·51-s − 4·57-s − 4·59-s − 11·67-s − 25·73-s + 3·75-s + 81-s − 20·83-s + 12·89-s − 2·97-s + 2·99-s + 6·107-s − 10·121-s + 6·123-s + 127-s + 10·129-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.485·17-s + 0.917·19-s − 3/5·25-s − 0.192·27-s − 0.348·33-s − 0.937·41-s − 1.52·43-s − 5/7·49-s + 0.280·51-s − 0.529·57-s − 0.520·59-s − 1.34·67-s − 2.92·73-s + 0.346·75-s + 1/9·81-s − 2.19·83-s + 1.27·89-s − 0.203·97-s + 0.201·99-s + 0.580·107-s − 0.909·121-s + 0.541·123-s + 0.0887·127-s + 0.880·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(373248\)    =    \(2^{9} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(23.7986\)
Root analytic conductor: \(2.20870\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 373248,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.11.ac_o
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.c_ba
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.ae_bh
23$C_2^2$ \( 1 - 27 T^{2} + p^{2} T^{4} \) 2.23.a_abb
29$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.29.a_bf
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.37.a_bd
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.g_de
43$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.43.k_cx
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.a_n
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.53.a_p
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.e_aw
61$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.61.a_dh
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.l_gg
71$C_2^2$ \( 1 - 51 T^{2} + p^{2} T^{4} \) 2.71.a_abz
73$C_2$$\times$$C_2$ \( ( 1 + 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.z_lo
79$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.79.a_de
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.u_iw
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.am_gw
97$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.c_gd
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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

−8.553812567950771800089119313879, −7.926918401477742368879889967098, −7.51073820975455251784392364931, −6.99744129939434586257429357413, −6.61605362885247922164960036588, −6.11905064377903532147069166154, −5.62022252015488082036982977050, −5.17610415798908488533529088989, −4.50208112838872631003147557771, −4.21436442352872031950225779131, −3.34390997471376549751511136793, −2.99418993510939899326900494562, −1.88446349242853262003918087061, −1.34726131444899479941773237384, 0, 1.34726131444899479941773237384, 1.88446349242853262003918087061, 2.99418993510939899326900494562, 3.34390997471376549751511136793, 4.21436442352872031950225779131, 4.50208112838872631003147557771, 5.17610415798908488533529088989, 5.62022252015488082036982977050, 6.11905064377903532147069166154, 6.61605362885247922164960036588, 6.99744129939434586257429357413, 7.51073820975455251784392364931, 7.926918401477742368879889967098, 8.553812567950771800089119313879

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