Properties

Label 4-1425e2-1.1-c1e2-0-7
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 + 10·11-s + 5·16-s + 2·19-s − 14·29-s + 6·31-s − 3·36-s − 4·41-s + 30·44-s + 14·49-s − 16·59-s + 26·61-s + 3·64-s + 20·71-s + 6·76-s + 30·79-s + 81-s − 6·89-s − 10·99-s + 20·109-s − 42·116-s + 53·121-s + 18·124-s + ⋯
L(s)  = 1  + 3/2·4-s − 1/3·9-s + 3.01·11-s + 5/4·16-s + 0.458·19-s − 2.59·29-s + 1.07·31-s − 1/2·36-s − 0.624·41-s + 4.52·44-s + 2·49-s − 2.08·59-s + 3.32·61-s + 3/8·64-s + 2.37·71-s + 0.688·76-s + 3.37·79-s + 1/9·81-s − 0.635·89-s − 1.00·99-s + 1.91·109-s − 3.89·116-s + 4.81·121-s + 1.61·124-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

\(L(1)\) \(\approx\) \(4.592689855\)
\(L(\frac12)\) \(\approx\) \(4.592689855\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.11.ak_bv
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.23.a_bj
29$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.29.o_ed
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.31.ag_ct
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.53.a_p
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.61.aba_lf
67$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \) 2.67.a_acb
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.71.au_ji
73$C_2^2$ \( 1 - 121 T^{2} + p^{2} T^{4} \) 2.73.a_aer
79$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.79.abe_ot
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.83.a_adh
89$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.89.g_hf
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

−9.557460192845431505409679246555, −9.433950508057088759521013006006, −8.987135602512739372813957843460, −8.646729102185458122404938221479, −8.037415097275334874870375248209, −7.67397528166117069568974678175, −7.06337598321541852452035596566, −6.95403381292559703873272870824, −6.55966073514767648836525814697, −5.99950589895923952911704346261, −5.99023774590868564719781978101, −5.22712650114632442028959764215, −4.74312491393989274482022457611, −3.82581376882715718118456099489, −3.73041901814392379807716180843, −3.44977661335701412246732694844, −2.34058184010867785513644931037, −2.22330858465016496922546400472, −1.41547976126058926601527485918, −0.946752145016121156955186093056, 0.946752145016121156955186093056, 1.41547976126058926601527485918, 2.22330858465016496922546400472, 2.34058184010867785513644931037, 3.44977661335701412246732694844, 3.73041901814392379807716180843, 3.82581376882715718118456099489, 4.74312491393989274482022457611, 5.22712650114632442028959764215, 5.99023774590868564719781978101, 5.99950589895923952911704346261, 6.55966073514767648836525814697, 6.95403381292559703873272870824, 7.06337598321541852452035596566, 7.67397528166117069568974678175, 8.037415097275334874870375248209, 8.646729102185458122404938221479, 8.987135602512739372813957843460, 9.433950508057088759521013006006, 9.557460192845431505409679246555

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