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Properties

Label 4-588e2-1.1-c1e2-0-20
Degree $4$
Conductor $345744$
Sign $1$
Analytic cond. $22.0449$
Root an. cond. $2.16684$
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  − 3·3-s + 6·5-s + 6·9-s − 18·15-s + 6·17-s + 17·25-s − 9·27-s + 14·37-s + 12·41-s + 8·43-s + 36·45-s + 6·47-s − 18·51-s − 6·59-s + 10·67-s − 51·75-s − 2·79-s + 9·81-s − 24·83-s + 36·85-s − 18·89-s − 18·101-s − 34·109-s − 42·111-s − 5·121-s − 36·123-s + 18·125-s + ⋯
L(s)  = 1  − 1.73·3-s + 2.68·5-s + 2·9-s − 4.64·15-s + 1.45·17-s + 17/5·25-s − 1.73·27-s + 2.30·37-s + 1.87·41-s + 1.21·43-s + 5.36·45-s + 0.875·47-s − 2.52·51-s − 0.781·59-s + 1.22·67-s − 5.88·75-s − 0.225·79-s + 81-s − 2.63·83-s + 3.90·85-s − 1.90·89-s − 1.79·101-s − 3.25·109-s − 3.98·111-s − 0.454·121-s − 3.24·123-s + 1.60·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(345744\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(22.0449\)
Root analytic conductor: \(2.16684\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 345744,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.003303258\)
\(L(\frac12)\) \(\approx\) \(2.003303258\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + p T + p T^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.19.a_abj
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.23.a_at
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.a_ach
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.37.ao_et
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \) 2.53.a_adb
59$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.59.g_ex
61$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.61.a_z
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.67.ak_gd
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.71.a_abi
73$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.73.a_b
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.79.c_gd
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.89.s_jz
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.97.a_afq
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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

−10.67839478268227198784945148446, −10.63830359910547817457388253588, −9.876990144961881343803408571985, −9.839388205096241273268225224864, −9.353742456793621866110243868330, −9.188060818182894902548537176039, −8.154910938720373893466177188646, −7.75809730195670918914825514617, −7.07081749686318347383350850786, −6.68557620828769834040664009990, −6.06005870446595182910509290212, −5.87231272390842427803038745803, −5.48464513632875650463490222693, −5.40713197550317710214437206010, −4.41003334955479520884473913263, −4.13975689088732262430549243718, −2.80382599225240817578804327978, −2.43104883311192426457390811041, −1.43085290930777930662093663830, −1.04010036322101432334451815703, 1.04010036322101432334451815703, 1.43085290930777930662093663830, 2.43104883311192426457390811041, 2.80382599225240817578804327978, 4.13975689088732262430549243718, 4.41003334955479520884473913263, 5.40713197550317710214437206010, 5.48464513632875650463490222693, 5.87231272390842427803038745803, 6.06005870446595182910509290212, 6.68557620828769834040664009990, 7.07081749686318347383350850786, 7.75809730195670918914825514617, 8.154910938720373893466177188646, 9.188060818182894902548537176039, 9.353742456793621866110243868330, 9.839388205096241273268225224864, 9.876990144961881343803408571985, 10.63830359910547817457388253588, 10.67839478268227198784945148446

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