Properties

Label 4-980e2-1.1-c1e2-0-16
Degree $4$
Conductor $960400$
Sign $-1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
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  − 2·4-s − 5·9-s + 4·16-s − 25-s − 6·29-s + 10·36-s − 4·37-s + 24·53-s − 8·64-s + 16·81-s + 2·100-s + 14·109-s + 12·113-s + 12·116-s − 13·121-s + 127-s + 131-s + 137-s + 139-s − 20·144-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + ⋯
L(s)  = 1  − 4-s − 5/3·9-s + 16-s − 1/5·25-s − 1.11·29-s + 5/3·36-s − 0.657·37-s + 3.29·53-s − 64-s + 16/9·81-s + 1/5·100-s + 1.34·109-s + 1.12·113-s + 1.11·116-s − 1.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5/3·144-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{960400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2$C_2$ \( 1 + p T^{2} \)
5$C_2$ \( 1 + T^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 193 T^{2} + p^{2} T^{4} \)
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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

−8.072531132972816642931699798914, −7.53353357299583671681693712478, −7.15190569768835151448318214982, −6.56112689904154859098848925311, −5.86914414726978577860225943043, −5.68831226185819751199078977061, −5.32640287451757670345317462029, −4.78851018489545068431737537985, −4.19012039377624132698273056315, −3.64622029282843089012072345623, −3.29982280143545223832366827708, −2.57503205376192440481912334986, −1.99586422105932837771956024974, −0.879757134568912941858458516589, 0, 0.879757134568912941858458516589, 1.99586422105932837771956024974, 2.57503205376192440481912334986, 3.29982280143545223832366827708, 3.64622029282843089012072345623, 4.19012039377624132698273056315, 4.78851018489545068431737537985, 5.32640287451757670345317462029, 5.68831226185819751199078977061, 5.86914414726978577860225943043, 6.56112689904154859098848925311, 7.15190569768835151448318214982, 7.53353357299583671681693712478, 8.072531132972816642931699798914

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