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Properties

Label 4-285e2-1.1-c1e2-0-2
Degree $4$
Conductor $81225$
Sign $1$
Analytic cond. $5.17897$
Root an. cond. $1.50855$
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  + 2·2-s − 2·3-s + 4-s − 2·5-s − 4·6-s + 4·7-s + 3·9-s − 4·10-s − 2·12-s + 8·13-s + 8·14-s + 4·15-s + 16-s − 8·17-s + 6·18-s + 2·19-s − 2·20-s − 8·21-s + 4·23-s + 3·25-s + 16·26-s − 4·27-s + 4·28-s − 4·29-s + 8·30-s − 4·31-s − 2·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 1.63·6-s + 1.51·7-s + 9-s − 1.26·10-s − 0.577·12-s + 2.21·13-s + 2.13·14-s + 1.03·15-s + 1/4·16-s − 1.94·17-s + 1.41·18-s + 0.458·19-s − 0.447·20-s − 1.74·21-s + 0.834·23-s + 3/5·25-s + 3.13·26-s − 0.769·27-s + 0.755·28-s − 0.742·29-s + 1.46·30-s − 0.718·31-s − 0.353·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(81225\)    =    \(3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(5.17897\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 81225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.082571214\)
\(L(\frac12)\) \(\approx\) \(2.082571214\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_d
7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
13$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bo
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bq
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_s
29$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_m
31$C_4$ \( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_ag
37$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_bo
41$D_{4}$ \( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_em
43$D_{4}$ \( 1 - 4 T + 88 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_dk
47$D_{4}$ \( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_du
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.59.a_bu
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.61.a_dm
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$D_{4}$ \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_di
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$D_{4}$ \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_du
89$D_{4}$ \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.89.e_u
97$C_2^2$ \( 1 + 176 T^{2} + p^{2} T^{4} \) 2.97.a_gu
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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

−11.94422651748311600022830476308, −11.52499868984857256751197083355, −11.10329910737413387041482926454, −11.07373818285000871114656988342, −10.76851606133144051168658296443, −9.895306648110656728687637429620, −8.927391779320616342134129825796, −8.780432841597661212279461053054, −8.165941696646063731713821037254, −7.59207325691769332581278216658, −6.96848280434509019490203487705, −6.56302654898553469937207572746, −5.74205691389279477949312162713, −5.48498184650769896560590443491, −4.67595605683700315639815054953, −4.61747116333653192293579041578, −3.82041776524910840198178033245, −3.61823027123933242237126070482, −2.08829373944119483464532167414, −1.06457276340249043064631230540, 1.06457276340249043064631230540, 2.08829373944119483464532167414, 3.61823027123933242237126070482, 3.82041776524910840198178033245, 4.61747116333653192293579041578, 4.67595605683700315639815054953, 5.48498184650769896560590443491, 5.74205691389279477949312162713, 6.56302654898553469937207572746, 6.96848280434509019490203487705, 7.59207325691769332581278216658, 8.165941696646063731713821037254, 8.780432841597661212279461053054, 8.927391779320616342134129825796, 9.895306648110656728687637429620, 10.76851606133144051168658296443, 11.07373818285000871114656988342, 11.10329910737413387041482926454, 11.52499868984857256751197083355, 11.94422651748311600022830476308

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