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Properties

Label 4-290e2-1.1-c1e2-0-26
Degree $4$
Conductor $84100$
Sign $-1$
Analytic cond. $5.36228$
Root an. cond. $1.52172$
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 + 5-s − 5·9-s − 6·11-s + 16-s + 20-s − 4·25-s − 2·29-s − 6·31-s − 5·36-s + 4·41-s − 6·44-s − 5·45-s − 10·49-s − 6·55-s − 16·61-s + 64-s + 4·71-s + 30·79-s + 80-s + 16·81-s − 20·89-s + 30·99-s − 4·100-s − 16·101-s + 10·109-s − 2·116-s + ⋯
L(s)  = 1  + 1/2·4-s + 0.447·5-s − 5/3·9-s − 1.80·11-s + 1/4·16-s + 0.223·20-s − 4/5·25-s − 0.371·29-s − 1.07·31-s − 5/6·36-s + 0.624·41-s − 0.904·44-s − 0.745·45-s − 1.42·49-s − 0.809·55-s − 2.04·61-s + 1/8·64-s + 0.474·71-s + 3.37·79-s + 0.111·80-s + 16/9·81-s − 2.11·89-s + 3.01·99-s − 2/5·100-s − 1.59·101-s + 0.957·109-s − 0.185·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84100\)    =    \(2^{2} \cdot 5^{2} \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(5.36228\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 84100,\ (\ :1/2, 1/2),\ -1)\)

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$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 - T + p T^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.a_z
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.43.a_abj
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.47.a_acx
53$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.53.a_ap
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.a_fa
79$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.79.abe_ot
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−9.633495492381883994739439375568, −8.926995527685451598137983295064, −8.389384753088059863163172626096, −7.81565944790560260026240611700, −7.71436185563727090632996745083, −6.89413781928069079756972828354, −6.11658999858863662469236119804, −5.89877989020180959127647474305, −5.25136790623537531107690201816, −4.96570329407044523914241607212, −3.82505096342084054880139925422, −3.07071881135695746551942332895, −2.58882766939603611448371560350, −1.89442567285374802225669499718, 0, 1.89442567285374802225669499718, 2.58882766939603611448371560350, 3.07071881135695746551942332895, 3.82505096342084054880139925422, 4.96570329407044523914241607212, 5.25136790623537531107690201816, 5.89877989020180959127647474305, 6.11658999858863662469236119804, 6.89413781928069079756972828354, 7.71436185563727090632996745083, 7.81565944790560260026240611700, 8.389384753088059863163172626096, 8.926995527685451598137983295064, 9.633495492381883994739439375568

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