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Properties

Label 4-450e2-1.1-c1e2-0-10
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 + 3·3-s + 3·6-s + 4·7-s − 8-s + 6·9-s − 3·11-s + 4·13-s + 4·14-s − 16-s − 6·17-s + 6·18-s − 8·19-s + 12·21-s − 3·22-s + 6·23-s − 3·24-s + 4·26-s + 9·27-s + 6·29-s − 8·31-s − 9·33-s − 6·34-s + 16·37-s − 8·38-s + 12·39-s + 6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1.22·6-s + 1.51·7-s − 0.353·8-s + 2·9-s − 0.904·11-s + 1.10·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s + 1.41·18-s − 1.83·19-s + 2.61·21-s − 0.639·22-s + 1.25·23-s − 0.612·24-s + 0.784·26-s + 1.73·27-s + 1.11·29-s − 1.43·31-s − 1.56·33-s − 1.02·34-s + 2.63·37-s − 1.29·38-s + 1.92·39-s + 0.937·41-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

\(L(1)\) \(\approx\) \(4.930325271\)
\(L(\frac12)\) \(\approx\) \(4.930325271\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ae_j
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2^2$ \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_d
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.17.g_br
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.29.ag_h
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bh
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.37.aq_fi
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_af
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_dt
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_w
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.73.abc_ne
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ap
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2^2$ \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.97.ah_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

−11.14314340690204897746785475618, −11.02870822182463731242825831331, −10.55519244422138762913497176099, −9.937741273579913439812830236781, −9.164694220557071644162259824268, −9.066275175637467104173195434364, −8.516292674807413646601919492850, −8.074169004599430264829075125345, −8.020053528427071814707162149594, −7.30172789688180291354122092982, −6.63661319606526328481181839060, −6.27593653342452062053381268112, −5.46079380721728476064883568850, −4.76714527921270499880497231625, −4.46505986533899908881375319689, −4.05473827716995837511463240104, −3.36781666777848659652600868573, −2.45413470281923141491536517104, −2.34412185611839136063214174642, −1.34955636937724162971654697395, 1.34955636937724162971654697395, 2.34412185611839136063214174642, 2.45413470281923141491536517104, 3.36781666777848659652600868573, 4.05473827716995837511463240104, 4.46505986533899908881375319689, 4.76714527921270499880497231625, 5.46079380721728476064883568850, 6.27593653342452062053381268112, 6.63661319606526328481181839060, 7.30172789688180291354122092982, 8.020053528427071814707162149594, 8.074169004599430264829075125345, 8.516292674807413646601919492850, 9.066275175637467104173195434364, 9.164694220557071644162259824268, 9.937741273579913439812830236781, 10.55519244422138762913497176099, 11.02870822182463731242825831331, 11.14314340690204897746785475618

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