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Properties

Label 4-170e2-1.1-c1e2-0-12
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s − 4·13-s − 16-s − 6·17-s − 8·19-s − 25-s + 4·26-s − 5·32-s + 6·34-s + 8·38-s + 2·43-s + 18·47-s − 8·49-s + 50-s + 4·52-s − 8·53-s − 16·59-s + 7·64-s + 6·67-s + 6·68-s + 8·76-s − 9·81-s − 26·83-s − 2·86-s − 18·94-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.10·13-s − 1/4·16-s − 1.45·17-s − 1.83·19-s − 1/5·25-s + 0.784·26-s − 0.883·32-s + 1.02·34-s + 1.29·38-s + 0.304·43-s + 2.62·47-s − 8/7·49-s + 0.141·50-s + 0.554·52-s − 1.09·53-s − 2.08·59-s + 7/8·64-s + 0.733·67-s + 0.727·68-s + 0.917·76-s − 81-s − 2.85·83-s − 0.215·86-s − 1.85·94-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: \(1\)
Selberg data: \((4,\ 28900,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.7.a_i
11$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.11.a_ai
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.e_o
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.i_bm
23$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.23.a_bo
29$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.29.a_be
31$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.31.a_ai
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.37.a_g
41$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.41.a_cg
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.ac_abi
47$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.47.as_gs
53$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.i_eo
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.q_gk
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.a_cg
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ag_dq
71$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \) 2.71.a_acu
73$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.73.a_abe
79$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \) 2.79.a_eq
83$C_2$$\times$$C_2$ \( ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.83.ba_mw
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.97.a_ade
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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

−10.21297280232623962707754863672, −9.708254981814360858292663699128, −9.202072821355533222558179266565, −8.662434285870896733895612797810, −8.423005787233979230346840942154, −7.53639490843311349121052775111, −7.27812164456876842814157434085, −6.46471330296411122639951342724, −5.93303168624518243897643367682, −4.94139277905990953926192582146, −4.48404698138439520044486747643, −3.99612885944062491515261866857, −2.68559110089798692482543213228, −1.85990320101749540748192083329, 0, 1.85990320101749540748192083329, 2.68559110089798692482543213228, 3.99612885944062491515261866857, 4.48404698138439520044486747643, 4.94139277905990953926192582146, 5.93303168624518243897643367682, 6.46471330296411122639951342724, 7.27812164456876842814157434085, 7.53639490843311349121052775111, 8.423005787233979230346840942154, 8.662434285870896733895612797810, 9.202072821355533222558179266565, 9.708254981814360858292663699128, 10.21297280232623962707754863672

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