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Properties

Label 4-290e2-1.1-c1e2-0-29
Degree $4$
Conductor $84100$
Sign $-1$
Analytic cond. $5.36228$
Root an. cond. $1.52172$
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  + 2-s + 2·3-s − 4-s − 2·5-s + 2·6-s − 3·8-s − 2·9-s − 2·10-s − 2·12-s − 4·15-s − 16-s − 2·18-s + 2·20-s − 6·24-s − 25-s − 10·27-s + 2·29-s − 4·30-s + 5·32-s + 2·36-s − 4·37-s + 6·40-s − 18·43-s + 4·45-s − 10·47-s − 2·48-s − 10·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 1.06·8-s − 2/3·9-s − 0.632·10-s − 0.577·12-s − 1.03·15-s − 1/4·16-s − 0.471·18-s + 0.447·20-s − 1.22·24-s − 1/5·25-s − 1.92·27-s + 0.371·29-s − 0.730·30-s + 0.883·32-s + 1/3·36-s − 0.657·37-s + 0.948·40-s − 2.74·43-s + 0.596·45-s − 1.45·47-s − 0.288·48-s − 1.42·49-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: yes
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$C_2$ \( 1 - T + p T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
29$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.3.ac_g
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
31$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.31.a_g
37$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.e_cw
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.41.a_ba
43$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.s_gk
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.47.k_dq
53$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.53.a_k
59$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.e_w
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.61.a_ba
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.83.a_k
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.89.a_o
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.ae_fe
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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

−9.449891785096614278356747588568, −8.762829961621822948720894592939, −8.414587251843563490092614595512, −8.083610984972066010171985886107, −7.76460406942667722047954302630, −6.78325596559468093746028558071, −6.42249890846583097469996365690, −5.65241399490463691461189947963, −5.03656855677052501563156974158, −4.62745717882541491468475126333, −3.69816465556431078308381615572, −3.40377485024070847574304979967, −2.97203791932766059943127967949, −1.96282224438940450408157738907, 0, 1.96282224438940450408157738907, 2.97203791932766059943127967949, 3.40377485024070847574304979967, 3.69816465556431078308381615572, 4.62745717882541491468475126333, 5.03656855677052501563156974158, 5.65241399490463691461189947963, 6.42249890846583097469996365690, 6.78325596559468093746028558071, 7.76460406942667722047954302630, 8.083610984972066010171985886107, 8.414587251843563490092614595512, 8.762829961621822948720894592939, 9.449891785096614278356747588568

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