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Properties

Label 4-416000-1.1-c1e2-0-16
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 − 4·9-s + 5·13-s − 6·17-s + 25-s − 6·29-s − 4·45-s − 14·49-s + 12·53-s + 2·61-s + 5·65-s + 7·81-s − 6·85-s + 6·89-s − 24·101-s − 4·109-s − 12·113-s − 20·117-s − 4·121-s + 125-s + 127-s + 131-s + 137-s + 139-s − 6·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.447·5-s − 4/3·9-s + 1.38·13-s − 1.45·17-s + 1/5·25-s − 1.11·29-s − 0.596·45-s − 2·49-s + 1.64·53-s + 0.256·61-s + 0.620·65-s + 7/9·81-s − 0.650·85-s + 0.635·89-s − 2.38·101-s − 0.383·109-s − 1.12·113-s − 1.84·117-s − 0.363·121-s + 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.498·145-s + 0.0819·149-s + 0.0813·151-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$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.g_bi
19$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.19.a_q
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.g_cg
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.43.a_acy
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.59.a_adc
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.ac_bq
67$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.67.a_ba
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.71.a_ec
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.83.a_acg
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ag_ec
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

−8.566741917343224085730634836387, −8.097824132697534456738344655883, −7.51703378040757846651698566869, −6.90211904959983832163348154497, −6.34221852988341277646913908593, −6.23596962589936671444651989580, −5.47478418976523825879715724916, −5.30827113578352840471627502094, −4.50848440168852603876823113425, −3.90524159353670016191437600504, −3.42690955195908553262574078563, −2.70546050896683527695415682624, −2.16567587921761779675385162953, −1.33284072610090064849709687162, 0, 1.33284072610090064849709687162, 2.16567587921761779675385162953, 2.70546050896683527695415682624, 3.42690955195908553262574078563, 3.90524159353670016191437600504, 4.50848440168852603876823113425, 5.30827113578352840471627502094, 5.47478418976523825879715724916, 6.23596962589936671444651989580, 6.34221852988341277646913908593, 6.90211904959983832163348154497, 7.51703378040757846651698566869, 8.097824132697534456738344655883, 8.566741917343224085730634836387

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