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Properties

Label 4-700e2-1.1-c1e2-0-10
Degree $4$
Conductor $490000$
Sign $-1$
Analytic cond. $31.2428$
Root an. cond. $2.36421$
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-s − 4-s − 3·8-s − 9-s − 6·13-s − 16-s + 4·17-s − 18-s − 6·26-s + 6·29-s + 5·32-s + 4·34-s + 36-s + 6·37-s − 6·41-s + 49-s + 6·52-s + 2·53-s + 6·58-s − 4·61-s + 7·64-s − 4·68-s + 3·72-s + 12·73-s + 6·74-s − 8·81-s − 6·82-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/3·9-s − 1.66·13-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.17·26-s + 1.11·29-s + 0.883·32-s + 0.685·34-s + 1/6·36-s + 0.986·37-s − 0.937·41-s + 1/7·49-s + 0.832·52-s + 0.274·53-s + 0.787·58-s − 0.512·61-s + 7/8·64-s − 0.485·68-s + 0.353·72-s + 1.40·73-s + 0.697·74-s − 8/9·81-s − 0.662·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(490000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(31.2428\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 490000,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
5 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.11.a_d
13$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.g_bi
17$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.17.ae_bd
19$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.19.a_l
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.31.a_k
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ag_cg
41$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.41.g_bb
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.47.a_cc
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.ac_du
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.59.a_as
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \) 2.67.a_aep
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \) 2.71.a_aeg
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.73.am_gb
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$C_2^2$ \( 1 - 87 T^{2} + p^{2} T^{4} \) 2.83.a_adj
89$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.89.s_hn
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−8.252818180470357764842845764436, −7.88168771386780471505655762516, −7.43591409041346529657793384057, −6.76640486847301517400298008736, −6.50853201149725989194514533363, −5.68132074281376792518268081312, −5.52034702675461869673387682345, −4.92725852691573408311308261574, −4.56464990608240063803500415493, −4.02198087305869023714899120853, −3.35637638885930627082458538435, −2.80549575878520939326902290746, −2.38407666571573695290426850338, −1.17001645422738999705577748510, 0, 1.17001645422738999705577748510, 2.38407666571573695290426850338, 2.80549575878520939326902290746, 3.35637638885930627082458538435, 4.02198087305869023714899120853, 4.56464990608240063803500415493, 4.92725852691573408311308261574, 5.52034702675461869673387682345, 5.68132074281376792518268081312, 6.50853201149725989194514533363, 6.76640486847301517400298008736, 7.43591409041346529657793384057, 7.88168771386780471505655762516, 8.252818180470357764842845764436

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