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Properties

Label 4-416000-1.1-c1e2-0-3
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·9-s − 3·13-s + 8·17-s + 25-s + 12·29-s − 8·37-s − 12·41-s + 2·45-s − 2·49-s + 8·53-s + 4·61-s + 3·65-s + 24·73-s − 5·81-s − 8·85-s − 12·89-s − 16·97-s − 4·101-s + 4·109-s − 8·113-s + 6·117-s + 10·121-s − 125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.447·5-s − 2/3·9-s − 0.832·13-s + 1.94·17-s + 1/5·25-s + 2.22·29-s − 1.31·37-s − 1.87·41-s + 0.298·45-s − 2/7·49-s + 1.09·53-s + 0.512·61-s + 0.372·65-s + 2.80·73-s − 5/9·81-s − 0.867·85-s − 1.27·89-s − 1.62·97-s − 0.398·101-s + 0.383·109-s − 0.752·113-s + 0.554·117-s + 0.909·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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: \(0\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.430043658\)
\(L(\frac12)\) \(\approx\) \(1.430043658\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ai_bu
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.i_di
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.m_dy
43$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.43.a_ack
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ai_di
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.67.a_bq
71$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.71.a_adu
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.73.ay_la
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.79.a_be
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.q_io
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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

−8.472185204453205089677187676094, −8.173285667836393536195579068934, −7.905321026799267165092596313049, −7.14272553366297499985588104544, −6.87244760561348829361207912365, −6.43735840736498996206227337043, −5.60236653493783690937614741213, −5.34445216544719044588276389950, −4.92798947435691028126045785847, −4.25761203392217673702774453546, −3.56460986099848623811247917846, −3.11569434555857186838028052898, −2.62178311172639077435678065113, −1.65959699787693501322506254898, −0.66888203914104658278403044439, 0.66888203914104658278403044439, 1.65959699787693501322506254898, 2.62178311172639077435678065113, 3.11569434555857186838028052898, 3.56460986099848623811247917846, 4.25761203392217673702774453546, 4.92798947435691028126045785847, 5.34445216544719044588276389950, 5.60236653493783690937614741213, 6.43735840736498996206227337043, 6.87244760561348829361207912365, 7.14272553366297499985588104544, 7.905321026799267165092596313049, 8.173285667836393536195579068934, 8.472185204453205089677187676094

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