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Properties

Label 4-50400-1.1-c1e2-0-5
Degree $4$
Conductor $50400$
Sign $-1$
Analytic cond. $3.21354$
Root an. cond. $1.33889$
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 − 6-s + 8-s − 2·9-s − 9·11-s − 12-s + 16-s − 3·17-s − 2·18-s − 9·22-s − 24-s − 5·25-s + 5·27-s + 32-s + 9·33-s − 3·34-s − 2·36-s + 2·43-s − 9·44-s − 48-s − 6·49-s − 5·50-s + 3·51-s − 3·53-s + 5·54-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 2.71·11-s − 0.288·12-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 1.91·22-s − 0.204·24-s − 25-s + 0.962·27-s + 0.176·32-s + 1.56·33-s − 0.514·34-s − 1/3·36-s + 0.304·43-s − 1.35·44-s − 0.144·48-s − 6/7·49-s − 0.707·50-s + 0.420·51-s − 0.412·53-s + 0.680·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(50400\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(3.21354\)
Root analytic conductor: \(1.33889\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 50400,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
5$C_2$ \( 1 + p T^{2} \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
good11$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.j_bo
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.13.a_b
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.19.a_h
23$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.23.a_t
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.31.a_ai
37$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.37.a_au
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.41.a_k
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.ac_g
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.53.d_ac
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.59.m_fp
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.c_bq
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.b_ek
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.71.a_fd
73$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.73.a_bi
79$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \) 2.79.a_acz
83$C_2^2$ \( 1 + 13 T^{2} + p^{2} T^{4} \) 2.83.a_n
89$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.a_bi
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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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

−10.12059001844186657814099026927, −9.412982910434611128519950270052, −8.712844652086050236521752428908, −8.090123905937888941274447462270, −7.77208083678586599586722035969, −7.24513406747853221292040232362, −6.42221814960599996502036872324, −5.92426286365934678101685449218, −5.48947963801567357395329044054, −4.89766574923806192612752429485, −4.53126052469121777595611523938, −3.39360363760954429911968066711, −2.77255327311124512550109493135, −2.10777736960006880882349937285, 0, 2.10777736960006880882349937285, 2.77255327311124512550109493135, 3.39360363760954429911968066711, 4.53126052469121777595611523938, 4.89766574923806192612752429485, 5.48947963801567357395329044054, 5.92426286365934678101685449218, 6.42221814960599996502036872324, 7.24513406747853221292040232362, 7.77208083678586599586722035969, 8.090123905937888941274447462270, 8.712844652086050236521752428908, 9.412982910434611128519950270052, 10.12059001844186657814099026927

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