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Properties

Label 4-170e2-1.1-c1e2-0-5
Degree $4$
Conductor $28900$
Sign $1$
Analytic cond. $1.84268$
Root an. cond. $1.16509$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 9-s + 4·13-s + 16-s − 2·19-s + 25-s + 36-s + 4·43-s + 2·49-s + 4·52-s − 12·53-s − 6·59-s + 64-s + 16·67-s − 2·76-s − 8·81-s + 24·83-s − 30·89-s + 100-s − 8·103-s + 4·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 144-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s + 1.10·13-s + 1/4·16-s − 0.458·19-s + 1/5·25-s + 1/6·36-s + 0.609·43-s + 2/7·49-s + 0.554·52-s − 1.64·53-s − 0.781·59-s + 1/8·64-s + 1.95·67-s − 0.229·76-s − 8/9·81-s + 2.63·83-s − 3.17·89-s + 1/10·100-s − 0.788·103-s + 0.369·117-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/12·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(28900\)    =    \(2^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1.84268\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 28900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.462666526\)
\(L(\frac12)\) \(\approx\) \(1.462666526\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.13.ae_v
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.c_d
23$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.23.a_abm
29$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.29.a_n
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.37.a_ac
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.a_dh
53$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.m_fd
59$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.g_dn
61$C_2^2$ \( 1 - 83 T^{2} + p^{2} T^{4} \) 2.61.a_adf
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.67.aq_gg
71$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.a_cj
73$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.73.a_acn
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.83.ay_ko
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.89.be_pn
97$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.97.a_acn
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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

−10.73080275021228721202801950022, −9.996295388120425479535305965760, −9.592851079033158131937135256055, −8.917012573242350578630681426897, −8.403918398159603562384424788176, −7.897313528378189830842386552504, −7.28249213460393726927234277397, −6.62914880419480688448651750637, −6.22921452238402282771489042052, −5.58554071710736424595539814718, −4.80793858846127517122320566264, −4.04344523923000326788296079894, −3.38703540801981907758862534539, −2.44614439449370505910873075156, −1.38219083895019499938283635482, 1.38219083895019499938283635482, 2.44614439449370505910873075156, 3.38703540801981907758862534539, 4.04344523923000326788296079894, 4.80793858846127517122320566264, 5.58554071710736424595539814718, 6.22921452238402282771489042052, 6.62914880419480688448651750637, 7.28249213460393726927234277397, 7.897313528378189830842386552504, 8.403918398159603562384424788176, 8.917012573242350578630681426897, 9.592851079033158131937135256055, 9.996295388120425479535305965760, 10.73080275021228721202801950022

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