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Properties

Label 4-416000-1.1-c1e2-0-2
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 − 13-s − 12·17-s + 25-s + 12·41-s − 2·45-s − 2·49-s + 8·61-s + 65-s − 5·81-s + 12·85-s + 24·97-s + 12·101-s − 4·109-s + 24·113-s − 2·117-s − 10·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + ⋯
L(s)  = 1  − 0.447·5-s + 2/3·9-s − 0.277·13-s − 2.91·17-s + 1/5·25-s + 1.87·41-s − 0.298·45-s − 2/7·49-s + 1.02·61-s + 0.124·65-s − 5/9·81-s + 1.30·85-s + 2.43·97-s + 1.19·101-s − 0.383·109-s + 2.25·113-s − 0.184·117-s − 0.909·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-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

\(L(1)\) \(\approx\) \(1.293721531\)
\(L(\frac12)\) \(\approx\) \(1.293721531\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
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_k
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.43.a_abi
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.67.a_aw
71$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.71.a_aba
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.83.a_by
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.97.ay_mw
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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

−8.601543427862549753349158222651, −8.270345914044786377843773671987, −7.59927473777358273976848169412, −7.22644245172039751049037099145, −6.89215938008072998397650272360, −6.28584800041885676361161702058, −6.03828637995512977342757899154, −5.11892204910453507790662246754, −4.71626962871176924930287980142, −4.22372253080111894196140174531, −3.93852063927242312703658615182, −3.04568451021755509890400992972, −2.34868952047324928306589563071, −1.87010125280108875930728636467, −0.61049734682347867945203662546, 0.61049734682347867945203662546, 1.87010125280108875930728636467, 2.34868952047324928306589563071, 3.04568451021755509890400992972, 3.93852063927242312703658615182, 4.22372253080111894196140174531, 4.71626962871176924930287980142, 5.11892204910453507790662246754, 6.03828637995512977342757899154, 6.28584800041885676361161702058, 6.89215938008072998397650272360, 7.22644245172039751049037099145, 7.59927473777358273976848169412, 8.270345914044786377843773671987, 8.601543427862549753349158222651

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