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Properties

Label 4-290e2-1.1-c1e2-0-16
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  + 4-s − 5-s − 7·7-s + 2·9-s + 7·13-s + 16-s − 20-s − 14·23-s − 4·25-s − 7·28-s − 3·29-s + 7·35-s + 2·36-s − 2·45-s + 25·49-s + 7·52-s − 6·59-s − 14·63-s + 64-s − 7·65-s − 7·67-s + 2·71-s − 80-s − 5·81-s − 7·83-s − 49·91-s − 14·92-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.447·5-s − 2.64·7-s + 2/3·9-s + 1.94·13-s + 1/4·16-s − 0.223·20-s − 2.91·23-s − 4/5·25-s − 1.32·28-s − 0.557·29-s + 1.18·35-s + 1/3·36-s − 0.298·45-s + 25/7·49-s + 0.970·52-s − 0.781·59-s − 1.76·63-s + 1/8·64-s − 0.868·65-s − 0.855·67-s + 0.237·71-s − 0.111·80-s − 5/9·81-s − 0.768·83-s − 5.13·91-s − 1.45·92-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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + T + p T^{2} \)
29$C_2$ \( 1 + 3 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
7$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.h_y
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.11.a_r
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.13.ah_bg
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \) 2.19.a_ap
23$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.o_dq
31$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.31.a_abl
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.a_k
41$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.41.a_ar
43$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.43.a_aq
47$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.47.a_cm
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.g_da
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.61.a_dm
67$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.h_ce
71$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.71.ac_fn
73$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.73.a_abo
79$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.79.a_acv
83$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.83.h_gk
89$C_2^2$ \( 1 + 67 T^{2} + p^{2} T^{4} \) 2.89.a_cp
97$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \) 2.97.a_abw
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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

−9.660100658567968204324701813729, −9.002008048548201992988511815157, −8.511013124507057028212206861356, −7.80667119530824861011439729129, −7.48473996488662278634028448467, −6.69652267213173409276676987231, −6.29153290880658865341251691550, −6.09412313988380258228065364248, −5.59127213005269533779459396938, −4.20522629294545523243752438633, −3.80573233165532672403728584288, −3.52238454385453353036123393326, −2.69426620368097067476190130213, −1.63491524877908082869354231479, 0, 1.63491524877908082869354231479, 2.69426620368097067476190130213, 3.52238454385453353036123393326, 3.80573233165532672403728584288, 4.20522629294545523243752438633, 5.59127213005269533779459396938, 6.09412313988380258228065364248, 6.29153290880658865341251691550, 6.69652267213173409276676987231, 7.48473996488662278634028448467, 7.80667119530824861011439729129, 8.511013124507057028212206861356, 9.002008048548201992988511815157, 9.660100658567968204324701813729

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