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Properties

Label 4-1425e2-1.1-c1e2-0-6
Degree $4$
Conductor $2030625$
Sign $1$
Analytic cond. $129.474$
Root an. cond. $3.37323$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s − 9-s + 8·11-s + 5·16-s + 2·19-s + 4·29-s − 3·36-s − 12·41-s + 24·44-s − 2·49-s − 8·59-s + 28·61-s + 3·64-s + 6·76-s − 32·79-s + 81-s + 12·89-s − 8·99-s − 20·101-s + 4·109-s + 12·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + ⋯
L(s)  = 1  + 3/2·4-s − 1/3·9-s + 2.41·11-s + 5/4·16-s + 0.458·19-s + 0.742·29-s − 1/2·36-s − 1.87·41-s + 3.61·44-s − 2/7·49-s − 1.04·59-s + 3.58·61-s + 3/8·64-s + 0.688·76-s − 3.60·79-s + 1/9·81-s + 1.27·89-s − 0.804·99-s − 1.99·101-s + 0.383·109-s + 1.11·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2030625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2030625\)    =    \(3^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(129.474\)
Root analytic conductor: \(3.37323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2030625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.262994841\)
\(L(\frac12)\) \(\approx\) \(4.262994841\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.37.a_abm
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.61.abc_mg
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.73.a_by
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−9.741590522265006566680975132210, −9.481958381483959191976041164347, −8.800574333300306112035418339529, −8.508003813654514723432627106782, −8.333729227939872460370805960175, −7.39527134350166960669800759626, −7.38064736380632376936104902292, −6.72414658981315508780859296814, −6.52381795228938403612505969484, −6.36195998140886769062882542580, −5.66039290577097033882020423551, −5.33334201585904165831097787261, −4.68512884197574406508055658656, −3.92429353251517714504702568086, −3.86597306384345309933285015784, −2.96265959827291455783799455168, −2.87463356098237036622553010111, −1.81657798657046413889704244335, −1.65140732910450831487516863839, −0.860936728894236267564706660240, 0.860936728894236267564706660240, 1.65140732910450831487516863839, 1.81657798657046413889704244335, 2.87463356098237036622553010111, 2.96265959827291455783799455168, 3.86597306384345309933285015784, 3.92429353251517714504702568086, 4.68512884197574406508055658656, 5.33334201585904165831097787261, 5.66039290577097033882020423551, 6.36195998140886769062882542580, 6.52381795228938403612505969484, 6.72414658981315508780859296814, 7.38064736380632376936104902292, 7.39527134350166960669800759626, 8.333729227939872460370805960175, 8.508003813654514723432627106782, 8.800574333300306112035418339529, 9.481958381483959191976041164347, 9.741590522265006566680975132210

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