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Properties

Label 4-290e2-1.1-c1e2-0-30
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 $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 4·5-s − 4·7-s − 6·9-s − 4·11-s + 2·13-s + 16-s − 4·19-s + 4·20-s − 4·23-s + 11·25-s + 4·28-s − 4·29-s − 12·31-s + 16·35-s + 6·36-s + 4·37-s − 14·41-s + 16·43-s + 4·44-s + 24·45-s + 8·47-s + 8·49-s − 2·52-s − 10·53-s + 16·55-s − 18·61-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.78·5-s − 1.51·7-s − 2·9-s − 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.917·19-s + 0.894·20-s − 0.834·23-s + 11/5·25-s + 0.755·28-s − 0.742·29-s − 2.15·31-s + 2.70·35-s + 36-s + 0.657·37-s − 2.18·41-s + 2.43·43-s + 0.603·44-s + 3.57·45-s + 1.16·47-s + 8/7·49-s − 0.277·52-s − 1.37·53-s + 2.15·55-s − 2.30·61-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: \(2\)
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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
29$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_i
11$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_i
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.ac_c
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_i
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_i
31$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_cu
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.41.o_du
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.47.ai_eg
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.k_by
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2^2$ \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.61.s_gg
67$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_i
71$C_2^2$ \( 1 - 126 T^{2} + p^{2} T^{4} \) 2.71.a_aew
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_cu
83$C_2^2$ \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.83.au_hs
89$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.89.k_by
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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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

−11.29350504275425458586483676460, −11.28361077516690844634380712789, −10.65085904842344974059940060652, −10.44193243276503848994519347738, −9.415487334978156944447885768054, −9.180443769432840749973446699761, −8.610067205663178047381953755043, −8.318825121095446519846183576943, −7.59244161785207272084637565585, −7.53277378580681981275827804664, −6.56387599557386505116075951862, −5.99983821188424238126523234101, −5.63053361175863494927545631900, −4.93404096209532875790445095631, −4.07147288496430489442324599044, −3.60505248466390001464676300835, −3.14703169295051535289863379557, −2.47266408504706734771069392238, 0, 0, 2.47266408504706734771069392238, 3.14703169295051535289863379557, 3.60505248466390001464676300835, 4.07147288496430489442324599044, 4.93404096209532875790445095631, 5.63053361175863494927545631900, 5.99983821188424238126523234101, 6.56387599557386505116075951862, 7.53277378580681981275827804664, 7.59244161785207272084637565585, 8.318825121095446519846183576943, 8.610067205663178047381953755043, 9.180443769432840749973446699761, 9.415487334978156944447885768054, 10.44193243276503848994519347738, 10.65085904842344974059940060652, 11.28361077516690844634380712789, 11.29350504275425458586483676460

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