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Properties

Label 4-1040e2-1.1-c1e2-0-64
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 9-s + 7·13-s − 9·17-s + 3·25-s + 3·29-s − 6·37-s + 6·41-s − 2·45-s − 8·49-s + 61-s − 14·65-s − 3·73-s − 8·81-s + 18·85-s − 12·89-s − 12·97-s − 9·101-s + 3·109-s + 9·113-s + 7·117-s + 16·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.894·5-s + 1/3·9-s + 1.94·13-s − 2.18·17-s + 3/5·25-s + 0.557·29-s − 0.986·37-s + 0.937·41-s − 0.298·45-s − 8/7·49-s + 0.128·61-s − 1.73·65-s − 0.351·73-s − 8/9·81-s + 1.95·85-s − 1.27·89-s − 1.21·97-s − 0.895·101-s + 0.287·109-s + 0.846·113-s + 0.647·117-s + 1.45·121-s − 0.357·125-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: \(1\)
Selberg data: \((4,\ 1081600,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 - 7 T + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.7.a_i
11$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.11.a_aq
17$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.j_ca
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.23.a_e
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.29.ad_bo
31$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.31.a_at
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.g_de
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.ag_de
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.43.a_cg
47$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.47.a_abo
53$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.a_dt
59$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.59.a_dc
61$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.61.ab_co
67$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.67.a_an
71$C_2^2$ \( 1 + 107 T^{2} + p^{2} T^{4} \) 2.71.a_ed
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.d_ai
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.83.a_ah
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.m_gw
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.m_ig
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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

−8.071336544190088110140753296604, −7.24753423310048476971114202616, −7.01567950370707920960186560489, −6.66422806309447469419266452942, −6.04814283754904228214325135600, −5.86126766661494313807303220316, −5.01348186319061502138678753182, −4.51685200252532945953641322425, −4.22735669691163524297793246373, −3.71530190467860307182401261798, −3.24158722031389631255919924709, −2.56408441706378219962171694405, −1.80092052931353088933378445381, −1.10370938172436550231985886095, 0, 1.10370938172436550231985886095, 1.80092052931353088933378445381, 2.56408441706378219962171694405, 3.24158722031389631255919924709, 3.71530190467860307182401261798, 4.22735669691163524297793246373, 4.51685200252532945953641322425, 5.01348186319061502138678753182, 5.86126766661494313807303220316, 6.04814283754904228214325135600, 6.66422806309447469419266452942, 7.01567950370707920960186560489, 7.24753423310048476971114202616, 8.071336544190088110140753296604

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