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Properties

Label 4-209952-1.1-c1e2-0-10
Degree $4$
Conductor $209952$
Sign $-1$
Analytic cond. $13.3867$
Root an. cond. $1.91279$
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 − 8-s − 6·11-s + 4·13-s + 16-s + 6·22-s − 3·23-s − 4·25-s − 4·26-s − 32-s + 7·37-s − 6·44-s + 3·46-s + 3·47-s − 4·49-s + 4·50-s + 4·52-s + 6·59-s − 17·61-s + 64-s − 9·71-s − 2·73-s − 7·74-s + 3·83-s + 6·88-s − 3·92-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s + 1.10·13-s + 1/4·16-s + 1.27·22-s − 0.625·23-s − 4/5·25-s − 0.784·26-s − 0.176·32-s + 1.15·37-s − 0.904·44-s + 0.442·46-s + 0.437·47-s − 4/7·49-s + 0.565·50-s + 0.554·52-s + 0.781·59-s − 2.17·61-s + 1/8·64-s − 1.06·71-s − 0.234·73-s − 0.813·74-s + 0.329·83-s + 0.639·88-s − 0.312·92-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(209952\)    =    \(2^{5} \cdot 3^{8}\)
Sign: $-1$
Analytic conductor: \(13.3867\)
Root analytic conductor: \(1.91279\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 209952,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.17.a_b
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.d_bc
29$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.29.a_au
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.37.ah_co
41$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.41.a_abj
43$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.43.a_abp
47$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.ad_cy
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.53.a_ac
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.59.ag_bu
61$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.r_gs
67$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.a_abj
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.71.j_fm
73$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.c_co
79$C_2^2$ \( 1 + 130 T^{2} + p^{2} T^{4} \) 2.79.a_fa
83$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.ad_ao
89$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.89.a_ar
97$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.ah_y
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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

−8.754313887860678585642650051942, −8.267847849839037571968174676048, −7.916975231556401887365348684940, −7.62418836369938413605395359390, −7.05429072900205640367199706026, −6.32838155909901323672634845899, −5.94195824079011379761831550892, −5.54860452776490384871384529180, −4.85099677725082134639423748388, −4.21690158411758781870218999027, −3.51856134792203120441725440813, −2.82005214973772042781175894824, −2.25577800181923052862308336290, −1.32996351238838463542713950058, 0, 1.32996351238838463542713950058, 2.25577800181923052862308336290, 2.82005214973772042781175894824, 3.51856134792203120441725440813, 4.21690158411758781870218999027, 4.85099677725082134639423748388, 5.54860452776490384871384529180, 5.94195824079011379761831550892, 6.32838155909901323672634845899, 7.05429072900205640367199706026, 7.62418836369938413605395359390, 7.916975231556401887365348684940, 8.267847849839037571968174676048, 8.754313887860678585642650051942

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