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Properties

Label 4-1040e2-1.1-c1e2-0-37
Degree $4$
Conductor $1081600$
Sign $1$
Analytic cond. $68.9637$
Root an. cond. $2.88174$
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  + 2·5-s − 4·9-s + 6·13-s + 4·17-s − 25-s + 6·37-s + 6·41-s − 8·45-s − 2·49-s + 10·53-s + 8·61-s + 12·65-s + 6·73-s + 7·81-s + 8·85-s − 16·89-s − 8·97-s − 24·101-s + 22·109-s + 8·113-s − 24·117-s − 14·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 4/3·9-s + 1.66·13-s + 0.970·17-s − 1/5·25-s + 0.986·37-s + 0.937·41-s − 1.19·45-s − 2/7·49-s + 1.37·53-s + 1.02·61-s + 1.48·65-s + 0.702·73-s + 7/9·81-s + 0.867·85-s − 1.69·89-s − 0.812·97-s − 2.38·101-s + 2.10·109-s + 0.752·113-s − 2.21·117-s − 1.27·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1081600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1081600\)    =    \(2^{8} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(68.9637\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.544593124\)
\(L(\frac12)\) \(\approx\) \(2.544593124\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.23.a_bg
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.ag_de
41$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ag_dm
43$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.43.a_dc
47$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.47.a_abu
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) 2.53.ak_ec
59$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.59.a_du
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.61.ai_es
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.ag_bi
79$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.79.a_aek
83$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.83.a_c
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.q_je
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.i_gs
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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

−8.038201148587239368939786663179, −7.896542078179330946942612146881, −7.13376651025622144307446672054, −6.65533223546090520974460944928, −6.17676990846627391965394008165, −5.82543781900637110613130219791, −5.52649638308215054263231193827, −5.24246366664265876187362669717, −4.28627583040497169970375916051, −3.94487366803648389136377333507, −3.29885614044111460437886702246, −2.82089107649376645599585354856, −2.26968596167663197319551350234, −1.48532116967127653952075589912, −0.76806841431114014802239967779, 0.76806841431114014802239967779, 1.48532116967127653952075589912, 2.26968596167663197319551350234, 2.82089107649376645599585354856, 3.29885614044111460437886702246, 3.94487366803648389136377333507, 4.28627583040497169970375916051, 5.24246366664265876187362669717, 5.52649638308215054263231193827, 5.82543781900637110613130219791, 6.17676990846627391965394008165, 6.65533223546090520974460944928, 7.13376651025622144307446672054, 7.896542078179330946942612146881, 8.038201148587239368939786663179

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