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Properties

Label 4-450e2-1.1-c1e2-0-6
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $12.9115$
Root an. cond. $1.89559$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·7-s − 8-s − 3·9-s − 4·13-s + 2·14-s − 16-s + 12·17-s − 3·18-s − 14·19-s − 4·26-s + 6·29-s + 10·31-s + 12·34-s − 4·37-s − 14·38-s − 9·41-s − 43-s + 6·47-s + 7·49-s + 24·53-s − 2·56-s + 6·58-s + 9·59-s + 4·61-s + 10·62-s − 6·63-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.755·7-s − 0.353·8-s − 9-s − 1.10·13-s + 0.534·14-s − 1/4·16-s + 2.91·17-s − 0.707·18-s − 3.21·19-s − 0.784·26-s + 1.11·29-s + 1.79·31-s + 2.05·34-s − 0.657·37-s − 2.27·38-s − 1.40·41-s − 0.152·43-s + 0.875·47-s + 49-s + 3.29·53-s − 0.267·56-s + 0.787·58-s + 1.17·59-s + 0.512·61-s + 1.27·62-s − 0.755·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(12.9115\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.124208492\)
\(L(\frac12)\) \(\approx\) \(2.124208492\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + p T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_ad
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_d
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.19.o_dj
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_cr
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_bo
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_al
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.53.ay_jq
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_w
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_abt
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.67.n_dy
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.73.ac_fr
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_acx
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.89.abe_pn
97$C_2^2$ \( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.97.ar_hk
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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

−11.65472466997933109799150388013, −10.57021510800665011692791231956, −10.47174748901983443790504414975, −10.27846725553952859944396915196, −9.625778802538188999736617137024, −8.716621750390633761285193056045, −8.712950734487194954332160072320, −8.020429779086631399442985567981, −7.939291179382744755355383843969, −7.00482223282437383387882383757, −6.66298818596799400334007289537, −5.90281402754987293962748121180, −5.67769869635210712620004877875, −4.89793932327320862923762674513, −4.80672030919431728779799726607, −3.91434509398070000055591273648, −3.49986925564631839877185963284, −2.51849519034468010214177899111, −2.27912390423606772154902281763, −0.834927215286127367792479834614, 0.834927215286127367792479834614, 2.27912390423606772154902281763, 2.51849519034468010214177899111, 3.49986925564631839877185963284, 3.91434509398070000055591273648, 4.80672030919431728779799726607, 4.89793932327320862923762674513, 5.67769869635210712620004877875, 5.90281402754987293962748121180, 6.66298818596799400334007289537, 7.00482223282437383387882383757, 7.939291179382744755355383843969, 8.020429779086631399442985567981, 8.712950734487194954332160072320, 8.716621750390633761285193056045, 9.625778802538188999736617137024, 10.27846725553952859944396915196, 10.47174748901983443790504414975, 10.57021510800665011692791231956, 11.65472466997933109799150388013

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