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Properties

Label 4-86400-1.1-c1e2-0-23
Degree $4$
Conductor $86400$
Sign $-1$
Analytic cond. $5.50893$
Root an. cond. $1.53202$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 15-s + 16-s − 18-s − 3·19-s + 20-s − 3·23-s + 24-s − 4·25-s − 27-s − 13·29-s + 30-s − 32-s + 36-s + 3·38-s − 40-s + 3·43-s + 45-s + 3·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.688·19-s + 0.223·20-s − 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s − 2.41·29-s + 0.182·30-s − 0.176·32-s + 1/6·36-s + 0.486·38-s − 0.158·40-s + 0.457·43-s + 0.149·45-s + 0.442·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86400\)    =    \(2^{7} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(5.50893\)
Root analytic conductor: \(1.53202\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 86400,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_1$ \( 1 + T \)
5$C_2$ \( 1 - T + p T^{2} \)
good7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.11.a_f
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.13.a_as
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.d_bi
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.d_bq
29$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.29.n_dw
31$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.31.a_bg
37$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.37.a_bp
41$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.41.a_acc
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.43.ad_dk
47$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.47.s_gp
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.j_ec
59$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \) 2.59.a_abx
61$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.61.a_ax
67$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.p_hc
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.ab_cs
73$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.73.am_gr
79$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.79.a_ack
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.83.a_ba
89$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \) 2.89.a_eu
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.ae_du
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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

−9.488508116022142522567121758527, −9.098504889599531682651349493401, −8.464207958085013711200527099444, −7.78793159490815791311120559755, −7.62405464012774177932176374876, −6.84883701755976447671101395475, −6.33771744127223914035700129551, −5.90213423364627850092735120642, −5.41621509922772428968222016010, −4.68967570047508704700511786422, −3.94759954382028462167055040267, −3.28473480709116034007044494996, −2.15723637421008685020540702177, −1.63963618836585869125685918219, 0, 1.63963618836585869125685918219, 2.15723637421008685020540702177, 3.28473480709116034007044494996, 3.94759954382028462167055040267, 4.68967570047508704700511786422, 5.41621509922772428968222016010, 5.90213423364627850092735120642, 6.33771744127223914035700129551, 6.84883701755976447671101395475, 7.62405464012774177932176374876, 7.78793159490815791311120559755, 8.464207958085013711200527099444, 9.098504889599531682651349493401, 9.488508116022142522567121758527

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