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Properties

Label 4-735e2-1.1-c1e2-0-12
Degree $4$
Conductor $540225$
Sign $1$
Analytic cond. $34.4452$
Root an. cond. $2.42260$
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  + 3·2-s + 4·4-s − 5-s + 3·8-s − 3·9-s − 3·10-s + 6·11-s + 3·16-s − 6·17-s − 9·18-s + 12·19-s − 4·20-s + 18·22-s − 3·23-s + 6·31-s + 6·32-s − 18·34-s − 12·36-s − 4·37-s + 36·38-s − 3·40-s − 6·41-s + 2·43-s + 24·44-s + 3·45-s − 9·46-s − 6·55-s + ⋯
L(s)  = 1  + 2.12·2-s + 2·4-s − 0.447·5-s + 1.06·8-s − 9-s − 0.948·10-s + 1.80·11-s + 3/4·16-s − 1.45·17-s − 2.12·18-s + 2.75·19-s − 0.894·20-s + 3.83·22-s − 0.625·23-s + 1.07·31-s + 1.06·32-s − 3.08·34-s − 2·36-s − 0.657·37-s + 5.83·38-s − 0.474·40-s − 0.937·41-s + 0.304·43-s + 3.61·44-s + 0.447·45-s − 1.32·46-s − 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(540225\)    =    \(3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(34.4452\)
Root analytic conductor: \(2.42260\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 540225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.656532900\)
\(L(\frac12)\) \(\approx\) \(5.656532900\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3$C_2$ \( 1 + p T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.2.ad_f
11$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_x
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_t
19$C_2^2$ \( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.19.am_cp
23$C_2^2$ \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ba
29$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.29.a_acd
31$C_2^2$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_br
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_av
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.43.ac_dj
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.53.a_cb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.61.aj_dk
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.67.an_dy
71$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.71.a_adq
73$C_2^2$ \( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.73.g_dh
79$C_2^2$ \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_gv
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.83.s_jn
89$C_2^2$ \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.89.d_adc
97$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.97.a_adi
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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

−10.79253791192828699264227887883, −10.30986572864683806260238432647, −9.687879286981908280421276478690, −9.283637333966890346433165406900, −9.004102997355472099512349656981, −8.179613566283504096843216732484, −8.167881731687850840018346598394, −7.31000313229571340733050202628, −6.74916628687139699289516607225, −6.66726941562789465227453115616, −5.85350390482400839288735701948, −5.65731500872816156413573041666, −5.07674130974125754058947313811, −4.67849208580994617409657087510, −4.15023812623342437303664274163, −3.70154989871732908996413724501, −3.33087541915833949707715243932, −2.83986354019931788790018960342, −1.93665306687718798642110202687, −0.933256382998182834583304606757, 0.933256382998182834583304606757, 1.93665306687718798642110202687, 2.83986354019931788790018960342, 3.33087541915833949707715243932, 3.70154989871732908996413724501, 4.15023812623342437303664274163, 4.67849208580994617409657087510, 5.07674130974125754058947313811, 5.65731500872816156413573041666, 5.85350390482400839288735701948, 6.66726941562789465227453115616, 6.74916628687139699289516607225, 7.31000313229571340733050202628, 8.167881731687850840018346598394, 8.179613566283504096843216732484, 9.004102997355472099512349656981, 9.283637333966890346433165406900, 9.687879286981908280421276478690, 10.30986572864683806260238432647, 10.79253791192828699264227887883

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