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Properties

Label 4-960e2-1.1-c1e2-0-34
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
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  + 2·3-s − 2·5-s + 2·7-s + 3·9-s + 6·11-s − 4·15-s − 10·17-s + 10·19-s + 4·21-s + 2·23-s − 25-s + 4·27-s + 10·29-s + 12·33-s − 4·35-s − 6·45-s + 10·47-s + 2·49-s − 20·51-s − 12·53-s − 12·55-s + 20·57-s + 10·59-s − 2·61-s + 6·63-s + 4·69-s + 10·73-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s + 1.80·11-s − 1.03·15-s − 2.42·17-s + 2.29·19-s + 0.872·21-s + 0.417·23-s − 1/5·25-s + 0.769·27-s + 1.85·29-s + 2.08·33-s − 0.676·35-s − 0.894·45-s + 1.45·47-s + 2/7·49-s − 2.80·51-s − 1.64·53-s − 1.61·55-s + 2.64·57-s + 1.30·59-s − 0.256·61-s + 0.755·63-s + 0.481·69-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.639878196\)
\(L(\frac12)\) \(\approx\) \(3.639878196\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_c
11$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_s
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.k_by
19$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.19.ak_by
23$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_c
29$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.29.ak_by
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.31.a_acg
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.47.ak_by
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_by
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.c_c
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.ak_by
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_c
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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

−10.13390554114030883258052801730, −9.470699761749704744869609264212, −9.426505913954670150834015433642, −8.901973742228694135986098614004, −8.668692728317559641115132535490, −8.130304995549508034832464742253, −7.86664830873683711510135873319, −7.33239343214184704711105346939, −6.82963197761125988078887888828, −6.73391383274102402782164596612, −6.08916887645060216933744696519, −5.31155363448019129218815294690, −4.58448892338414335863802584597, −4.57927297839479369730202278074, −3.82672208139228783270265163502, −3.61943083380974864244881459489, −2.84823519295779480135423030485, −2.35220409108836255471492392404, −1.52283366765786604601504134874, −0.933276557439396517521425656918, 0.933276557439396517521425656918, 1.52283366765786604601504134874, 2.35220409108836255471492392404, 2.84823519295779480135423030485, 3.61943083380974864244881459489, 3.82672208139228783270265163502, 4.57927297839479369730202278074, 4.58448892338414335863802584597, 5.31155363448019129218815294690, 6.08916887645060216933744696519, 6.73391383274102402782164596612, 6.82963197761125988078887888828, 7.33239343214184704711105346939, 7.86664830873683711510135873319, 8.130304995549508034832464742253, 8.668692728317559641115132535490, 8.901973742228694135986098614004, 9.426505913954670150834015433642, 9.470699761749704744869609264212, 10.13390554114030883258052801730

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