Properties

Label 4-416000-1.1-c1e2-0-15
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 − 3·13-s − 8·17-s + 25-s − 6·41-s − 4·45-s + 2·49-s + 12·61-s + 3·65-s + 4·73-s + 7·81-s + 8·85-s − 24·89-s + 18·97-s − 8·109-s − 28·113-s − 12·117-s + 6·121-s − 125-s + ⋯
L(s)  = 1  − 0.447·5-s + 4/3·9-s − 0.832·13-s − 1.94·17-s + 1/5·25-s − 0.937·41-s − 0.596·45-s + 2/7·49-s + 1.53·61-s + 0.372·65-s + 0.468·73-s + 7/9·81-s + 0.867·85-s − 2.54·89-s + 1.82·97-s − 0.766·109-s − 2.63·113-s − 1.10·117-s + 6/11·121-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$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
17$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.i_bu
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.23.a_ai
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.31.a_aw
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.g_co
43$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.43.a_abo
47$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.47.a_acw
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.59.a_abi
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.am_fm
67$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.67.a_de
71$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.71.a_du
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.ae_fe
79$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.79.a_ack
83$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.83.a_eg
89$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.y_mg
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.97.as_kg
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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

−8.504601103587961492517261555423, −7.84970394995994472817677922044, −7.42590299582404244046079235144, −7.02835910303777383867178520589, −6.62467415360465971384832752755, −6.32152630956903243867249403015, −5.34839914190875342394402138700, −5.04413196788326631738480636020, −4.39358314706614676740148173507, −4.13193510275735343204243369949, −3.55336441854810177608390128843, −2.62523999312601735491514756380, −2.15772516264223377605504437669, −1.29843321148078358345430763094, 0, 1.29843321148078358345430763094, 2.15772516264223377605504437669, 2.62523999312601735491514756380, 3.55336441854810177608390128843, 4.13193510275735343204243369949, 4.39358314706614676740148173507, 5.04413196788326631738480636020, 5.34839914190875342394402138700, 6.32152630956903243867249403015, 6.62467415360465971384832752755, 7.02835910303777383867178520589, 7.42590299582404244046079235144, 7.84970394995994472817677922044, 8.504601103587961492517261555423

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