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Properties

Label 4-960e2-1.1-c1e2-0-52
Degree $4$
Conductor $921600$
Sign $-1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
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  + 2·3-s − 4·7-s + 9-s − 4·13-s − 8·21-s + 25-s − 4·27-s + 8·31-s + 12·37-s − 8·39-s + 4·43-s − 2·49-s + 12·61-s − 4·63-s + 12·67-s − 12·73-s + 2·75-s − 16·79-s − 11·81-s + 16·91-s + 16·93-s − 12·97-s + 4·103-s − 20·109-s + 24·111-s − 4·117-s + 10·121-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.51·7-s + 1/3·9-s − 1.10·13-s − 1.74·21-s + 1/5·25-s − 0.769·27-s + 1.43·31-s + 1.97·37-s − 1.28·39-s + 0.609·43-s − 2/7·49-s + 1.53·61-s − 0.503·63-s + 1.46·67-s − 1.40·73-s + 0.230·75-s − 1.80·79-s − 1.22·81-s + 1.67·91-s + 1.65·93-s − 1.21·97-s + 0.394·103-s − 1.91·109-s + 2.27·111-s − 0.369·117-s + 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.e_o
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.23.a_aw
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ai_ck
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.37.am_dq
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.ae_ba
47$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.47.a_acs
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.a_adm
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.am_dq
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.am_ec
71$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.71.a_aco
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.m_eo
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.q_hy
83$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.83.a_ack
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.m_gk
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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

−8.137350963425156938143193081097, −7.54657663643855675749758962651, −7.02041610970919015682094892807, −6.80777200741494364431609064720, −6.14673126948954623765411910247, −5.87435664600668262606628281242, −5.22559694129132304125964987199, −4.56427235009010419370928876275, −4.17040177260755040988225508618, −3.55716240889162626911849324748, −3.03884925268646366677940027816, −2.59633589275285270307640031713, −2.33723627402858069538190940698, −1.14921338945524482167036855350, 0, 1.14921338945524482167036855350, 2.33723627402858069538190940698, 2.59633589275285270307640031713, 3.03884925268646366677940027816, 3.55716240889162626911849324748, 4.17040177260755040988225508618, 4.56427235009010419370928876275, 5.22559694129132304125964987199, 5.87435664600668262606628281242, 6.14673126948954623765411910247, 6.80777200741494364431609064720, 7.02041610970919015682094892807, 7.54657663643855675749758962651, 8.137350963425156938143193081097

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