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Properties

Label 4-1320e2-1.1-c1e2-0-51
Degree $4$
Conductor $1742400$
Sign $-1$
Analytic cond. $111.096$
Root an. cond. $3.24657$
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 + 3·5-s − 2·9-s − 2·11-s − 3·15-s − 4·23-s + 4·25-s + 5·27-s − 5·31-s + 2·33-s + 6·37-s − 6·45-s − 6·47-s + 10·49-s + 5·53-s − 6·55-s − 6·59-s + 7·67-s + 4·69-s − 7·71-s − 4·75-s + 81-s + 13·89-s + 5·93-s + 28·97-s + 4·99-s − 7·103-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 2/3·9-s − 0.603·11-s − 0.774·15-s − 0.834·23-s + 4/5·25-s + 0.962·27-s − 0.898·31-s + 0.348·33-s + 0.986·37-s − 0.894·45-s − 0.875·47-s + 10/7·49-s + 0.686·53-s − 0.809·55-s − 0.781·59-s + 0.855·67-s + 0.481·69-s − 0.830·71-s − 0.461·75-s + 1/9·81-s + 1.37·89-s + 0.518·93-s + 2.84·97-s + 0.402·99-s − 0.689·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1742400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(111.096\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1742400,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
11$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.a_r
17$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.17.a_i
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.e_bu
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.f_cq
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.37.ag_cw
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.41.a_c
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.47.g_dv
53$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) 2.53.af_ec
59$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.g_ew
61$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.61.a_acv
67$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.ah_o
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.h_w
73$C_2^2$ \( 1 - 69 T^{2} + p^{2} T^{4} \) 2.73.a_acr
79$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.79.a_ek
83$C_2^2$ \( 1 - 121 T^{2} + p^{2} T^{4} \) 2.83.a_aer
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.89.an_ig
97$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.97.abc_ow
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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

−7.51515793913511756630524359722, −7.24552142899280262554623403459, −6.49927934915805437801475102058, −6.22643770966893595214823620399, −5.94411523242652267267775321418, −5.48701117724538981355717252548, −5.10674837259636542814978370388, −4.78113628716589885422144688761, −3.99120132485445483910941867880, −3.56024692584848914413275474713, −2.71721447307338068828422026441, −2.45489459914752010840607708069, −1.84610986203243854483088274610, −1.04691632933745722708988300066, 0, 1.04691632933745722708988300066, 1.84610986203243854483088274610, 2.45489459914752010840607708069, 2.71721447307338068828422026441, 3.56024692584848914413275474713, 3.99120132485445483910941867880, 4.78113628716589885422144688761, 5.10674837259636542814978370388, 5.48701117724538981355717252548, 5.94411523242652267267775321418, 6.22643770966893595214823620399, 6.49927934915805437801475102058, 7.24552142899280262554623403459, 7.51515793913511756630524359722

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