Properties

Label 4-578e2-1.1-c1e2-0-2
Degree $4$
Conductor $334084$
Sign $1$
Analytic cond. $21.3014$
Root an. cond. $2.14833$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s + 2·9-s + 4·13-s + 5·16-s − 4·18-s + 8·19-s + 10·25-s − 8·26-s − 6·32-s + 6·36-s − 16·38-s − 16·43-s − 2·49-s − 20·50-s + 12·52-s + 12·53-s + 7·64-s + 16·67-s − 8·72-s + 24·76-s − 5·81-s + 32·86-s − 12·89-s + 4·98-s + 30·100-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2/3·9-s + 1.10·13-s + 5/4·16-s − 0.942·18-s + 1.83·19-s + 2·25-s − 1.56·26-s − 1.06·32-s + 36-s − 2.59·38-s − 2.43·43-s − 2/7·49-s − 2.82·50-s + 1.66·52-s + 1.64·53-s + 7/8·64-s + 1.95·67-s − 0.942·72-s + 2.75·76-s − 5/9·81-s + 3.45·86-s − 1.27·89-s + 0.404·98-s + 3·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 334084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(334084\)    =    \(2^{2} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(21.3014\)
Root analytic conductor: \(2.14833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 334084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.188881690\)
\(L(\frac12)\) \(\approx\) \(1.188881690\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
17 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−10.64896913563247714245311646407, −10.56793271005754251982042105805, −9.960876811290948746888204112180, −9.563932990430062457679054238899, −9.341239075667857016820254525143, −8.615096060217754910464155159115, −8.371326909021417023397777812427, −8.104149322960137769980037775363, −7.27094322719512629168055370326, −6.99155221989097526442870355040, −6.79379756635244739325087744273, −6.07441971692121125140075145350, −5.48806145679598763124225874359, −5.04441199958066535538761740686, −4.31721149977573346282347376739, −3.31409727277292343670850099827, −3.29287786882727944161359283658, −2.26481250631177369319880629320, −1.38660847017651924887315861907, −0.903121005584487045429289479848, 0.903121005584487045429289479848, 1.38660847017651924887315861907, 2.26481250631177369319880629320, 3.29287786882727944161359283658, 3.31409727277292343670850099827, 4.31721149977573346282347376739, 5.04441199958066535538761740686, 5.48806145679598763124225874359, 6.07441971692121125140075145350, 6.79379756635244739325087744273, 6.99155221989097526442870355040, 7.27094322719512629168055370326, 8.104149322960137769980037775363, 8.371326909021417023397777812427, 8.615096060217754910464155159115, 9.341239075667857016820254525143, 9.563932990430062457679054238899, 9.960876811290948746888204112180, 10.56793271005754251982042105805, 10.64896913563247714245311646407

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