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Properties

Label 4-456300-1.1-c1e2-0-3
Degree $4$
Conductor $456300$
Sign $-1$
Analytic cond. $29.0940$
Root an. cond. $2.32247$
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  − 3-s − 4-s + 9-s + 12-s − 2·13-s + 16-s − 25-s − 27-s − 36-s + 2·39-s − 8·43-s − 48-s + 2·49-s + 2·52-s + 20·61-s − 64-s + 75-s − 16·79-s + 81-s + 100-s − 8·103-s + 108-s − 2·117-s + 22·121-s + 127-s + 8·129-s + 131-s + ⋯
L(s)  = 1  − 0.577·3-s − 1/2·4-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 1/5·25-s − 0.192·27-s − 1/6·36-s + 0.320·39-s − 1.21·43-s − 0.144·48-s + 2/7·49-s + 0.277·52-s + 2.56·61-s − 1/8·64-s + 0.115·75-s − 1.80·79-s + 1/9·81-s + 1/10·100-s − 0.788·103-s + 0.0962·108-s − 0.184·117-s + 2·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(456300\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(29.0940\)
Root analytic conductor: \(2.32247\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 456300,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( 1 + T \)
5$C_2$ \( 1 + T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.41.a_abu
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.89.a_fq
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−8.426961667900432963481636187504, −7.919721723599843316877272792267, −7.34084816164363712403210260669, −7.00261963424523161274508583804, −6.49072089071247518053957852779, −5.99731292871378330820606436826, −5.31474000512286357557971750672, −5.21707378684093067268379875848, −4.50551760684476943665010993257, −4.04867066954241061979433580878, −3.48845598466258731448671350790, −2.74213439026381835441408934743, −2.02117350733386732368742541545, −1.10999263120959718375988906834, 0, 1.10999263120959718375988906834, 2.02117350733386732368742541545, 2.74213439026381835441408934743, 3.48845598466258731448671350790, 4.04867066954241061979433580878, 4.50551760684476943665010993257, 5.21707378684093067268379875848, 5.31474000512286357557971750672, 5.99731292871378330820606436826, 6.49072089071247518053957852779, 7.00261963424523161274508583804, 7.34084816164363712403210260669, 7.919721723599843316877272792267, 8.426961667900432963481636187504

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