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Properties

Label 4-416000-1.1-c1e2-0-18
Degree $4$
Conductor $416000$
Sign $-1$
Analytic cond. $26.5245$
Root an. cond. $2.26940$
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  + 5-s − 9-s − 13-s − 3·17-s + 25-s − 3·29-s + 3·37-s + 9·41-s − 45-s − 5·49-s − 12·53-s − 28·61-s − 65-s − 6·73-s − 8·81-s − 3·85-s − 6·89-s − 18·97-s − 3·101-s + 32·109-s − 21·113-s + 117-s − 121-s + 125-s + ⋯
L(s)  = 1  + 0.447·5-s − 1/3·9-s − 0.277·13-s − 0.727·17-s + 1/5·25-s − 0.557·29-s + 0.493·37-s + 1.40·41-s − 0.149·45-s − 5/7·49-s − 1.64·53-s − 3.58·61-s − 0.124·65-s − 0.702·73-s − 8/9·81-s − 0.325·85-s − 0.635·89-s − 1.82·97-s − 0.298·101-s + 3.06·109-s − 1.97·113-s + 0.0924·117-s − 0.0909·121-s + 0.0894·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(416000\)    =    \(2^{8} \cdot 5^{3} \cdot 13\)
Sign: $-1$
Analytic conductor: \(26.5245\)
Root analytic conductor: \(2.26940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$ \( 1 - T \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.7.a_f
11$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.11.a_b
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.19.a_abd
23$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.23.a_aq
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.d_cg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ad_cm
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.41.aj_dw
43$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.43.a_r
47$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \) 2.47.a_abx
53$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.m_fd
59$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.59.a_ar
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.61.bc_mg
67$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.67.a_o
71$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \) 2.71.a_aem
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.g_fy
79$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.79.a_acn
83$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.83.a_abo
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.g_ec
97$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.97.s_kl
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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

−8.422506382245165335677359176094, −7.88138839565620278609613296188, −7.55735042066306993088375804621, −7.01738806095567732471871477564, −6.48034955127513946981463571981, −5.98531844949415596370180491059, −5.75145236239952441206871769496, −5.00914816673734889717781297779, −4.50041305445817230188065122167, −4.16494990045116484284121229189, −3.17034870307914656639354212008, −2.87609408584538825537414892469, −2.06428958464620599014039653654, −1.38737573643414769285897528071, 0, 1.38737573643414769285897528071, 2.06428958464620599014039653654, 2.87609408584538825537414892469, 3.17034870307914656639354212008, 4.16494990045116484284121229189, 4.50041305445817230188065122167, 5.00914816673734889717781297779, 5.75145236239952441206871769496, 5.98531844949415596370180491059, 6.48034955127513946981463571981, 7.01738806095567732471871477564, 7.55735042066306993088375804621, 7.88138839565620278609613296188, 8.422506382245165335677359176094

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