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Properties

Label 4-245e2-1.1-c1e2-0-12
Degree $4$
Conductor $60025$
Sign $1$
Analytic cond. $3.82724$
Root an. cond. $1.39869$
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 − 9-s + 4·10-s + 4·11-s + 2·12-s − 4·13-s + 4·15-s + 16-s + 4·17-s − 2·18-s + 2·20-s + 8·22-s − 2·23-s + 3·25-s − 8·26-s − 6·27-s − 2·29-s + 8·30-s − 12·31-s − 2·32-s + 8·33-s + 8·34-s − 36-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s − 1/3·9-s + 1.26·10-s + 1.20·11-s + 0.577·12-s − 1.10·13-s + 1.03·15-s + 1/4·16-s + 0.970·17-s − 0.471·18-s + 0.447·20-s + 1.70·22-s − 0.417·23-s + 3/5·25-s − 1.56·26-s − 1.15·27-s − 0.371·29-s + 1.46·30-s − 2.15·31-s − 0.353·32-s + 1.39·33-s + 1.37·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60025\)    =    \(5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(3.82724\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 60025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.200436669\)
\(L(\frac12)\) \(\approx\) \(4.200436669\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good2$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_d
3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_f
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_s
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_w
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_be
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$D_{4}$ \( 1 + 2 T + 45 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bt
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.29.c_ch
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$D_{4}$ \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_dv
43$D_{4}$ \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_ef
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$D_{4}$ \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.53.i_ek
59$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_ck
61$D_{4}$ \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_ch
67$D_{4}$ \( 1 - 22 T + 253 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.67.aw_jt
71$D_{4}$ \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_di
73$D_{4}$ \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_fm
79$D_{4}$ \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.79.ay_li
83$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_f
89$D_{4}$ \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_fz
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_hq
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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

−12.37657961323749241891053169558, −12.21738619107954657487750275838, −11.59140178287151689150463252320, −10.90182647674348249924956185325, −10.51927318717257616782224371613, −9.680104268635009568238281529487, −9.381054512580261451282804578375, −9.128517205586172767917323920038, −8.519614347591273236320156022142, −7.77168289287107728268345394436, −7.46214629350117722925744398268, −6.69074264906189113526561940800, −6.06758895217091148698720327366, −5.47053587624388605004895475256, −5.14890638880266816814320518669, −4.41142485206502083151230136290, −3.52003517756079828896625081031, −3.47719629566148823841163821424, −2.42385574941462730227162082321, −1.75970914726115654218644943864, 1.75970914726115654218644943864, 2.42385574941462730227162082321, 3.47719629566148823841163821424, 3.52003517756079828896625081031, 4.41142485206502083151230136290, 5.14890638880266816814320518669, 5.47053587624388605004895475256, 6.06758895217091148698720327366, 6.69074264906189113526561940800, 7.46214629350117722925744398268, 7.77168289287107728268345394436, 8.519614347591273236320156022142, 9.128517205586172767917323920038, 9.381054512580261451282804578375, 9.680104268635009568238281529487, 10.51927318717257616782224371613, 10.90182647674348249924956185325, 11.59140178287151689150463252320, 12.21738619107954657487750275838, 12.37657961323749241891053169558

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