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Properties

Label 4-2178e2-1.1-c1e2-0-26
Degree $4$
Conductor $4743684$
Sign $-1$
Analytic cond. $302.461$
Root an. cond. $4.17030$
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  + 4-s + 2·5-s + 16-s + 2·20-s − 7·25-s − 16·31-s − 10·37-s − 8·47-s + 2·49-s + 10·53-s + 64-s + 24·67-s + 32·71-s + 2·80-s + 34·89-s − 10·97-s − 7·100-s + 8·103-s − 6·113-s − 16·124-s − 26·125-s + 127-s + 131-s + 137-s + 139-s − 10·148-s + 149-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.894·5-s + 1/4·16-s + 0.447·20-s − 7/5·25-s − 2.87·31-s − 1.64·37-s − 1.16·47-s + 2/7·49-s + 1.37·53-s + 1/8·64-s + 2.93·67-s + 3.79·71-s + 0.223·80-s + 3.60·89-s − 1.01·97-s − 0.699·100-s + 0.788·103-s − 0.564·113-s − 1.43·124-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.821·148-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4743684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4743684\)    =    \(2^{2} \cdot 3^{4} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(302.461\)
Root analytic conductor: \(4.17030\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 4743684,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
11 \( 1 \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.a_ap
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.29.a_j
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.37.k_dv
41$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.41.a_abn
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.47.i_eg
53$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.53.ak_fb
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.71.abg_pi
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.a_dy
89$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \) 2.89.abi_rz
97$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.97.k_il
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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

−7.26634431715188481710062362858, −6.57075345351642095686100440124, −6.44934005324156407715983783944, −5.92644738805241385436272478186, −5.31913862642412009263974020553, −5.31461075754579465246483014136, −4.91383230445445327227957950722, −3.84672567348487089358465749426, −3.70577819101418826144462093465, −3.48639974405348834102831799643, −2.45043366078956231065795739764, −2.04050454327404296238699132645, −1.94840976863843280215220992485, −1.04136613544003206830324521562, 0, 1.04136613544003206830324521562, 1.94840976863843280215220992485, 2.04050454327404296238699132645, 2.45043366078956231065795739764, 3.48639974405348834102831799643, 3.70577819101418826144462093465, 3.84672567348487089358465749426, 4.91383230445445327227957950722, 5.31461075754579465246483014136, 5.31913862642412009263974020553, 5.92644738805241385436272478186, 6.44934005324156407715983783944, 6.57075345351642095686100440124, 7.26634431715188481710062362858

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