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Properties

Label 4-1323e2-1.1-c1e2-0-11
Degree $4$
Conductor $1750329$
Sign $-1$
Analytic cond. $111.602$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4-s − 8·8-s − 10·11-s − 7·16-s − 20·22-s − 2·23-s − 25-s + 4·29-s + 14·32-s + 6·37-s + 12·43-s + 10·44-s − 4·46-s − 2·50-s + 16·53-s + 8·58-s + 35·64-s − 28·67-s + 14·71-s + 12·74-s − 20·79-s + 24·86-s + 80·88-s + 2·92-s + 100-s + 32·106-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 2.82·8-s − 3.01·11-s − 7/4·16-s − 4.26·22-s − 0.417·23-s − 1/5·25-s + 0.742·29-s + 2.47·32-s + 0.986·37-s + 1.82·43-s + 1.50·44-s − 0.589·46-s − 0.282·50-s + 2.19·53-s + 1.05·58-s + 35/8·64-s − 3.42·67-s + 1.66·71-s + 1.39·74-s − 2.25·79-s + 2.58·86-s + 8.52·88-s + 0.208·92-s + 1/10·100-s + 3.10·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1750329\)    =    \(3^{6} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(111.602\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1750329,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.2.ac_f
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.11.k_bv
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.19.a_bd
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.23.c_bv
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.37.ag_df
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.a_b
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.53.aq_go
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.a_de
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.67.bc_ms
71$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.71.ao_hj
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.a_gn
97$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.a_by
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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

−7.51628200491268451704050962091, −7.39691755528068449061259570564, −6.47884273436117871307974498700, −5.96466062289959274270681070046, −5.68667112029587934302581219019, −5.41389437444540368847999455753, −4.97324351018055834490293383395, −4.37690537415494649135748951679, −4.37621733692366770303800440112, −3.62230602066129289359790734359, −2.95504645610525949403863409430, −2.73512468913067121254189272250, −2.26481617188159673197543529003, −0.77022560857539424923599740084, 0, 0.77022560857539424923599740084, 2.26481617188159673197543529003, 2.73512468913067121254189272250, 2.95504645610525949403863409430, 3.62230602066129289359790734359, 4.37621733692366770303800440112, 4.37690537415494649135748951679, 4.97324351018055834490293383395, 5.41389437444540368847999455753, 5.68667112029587934302581219019, 5.96466062289959274270681070046, 6.47884273436117871307974498700, 7.39691755528068449061259570564, 7.51628200491268451704050962091

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