Properties

Label 4-676000-1.1-c1e2-0-41
Degree $4$
Conductor $676000$
Sign $-1$
Analytic cond. $43.1023$
Root an. cond. $2.56227$
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 − 5-s + 8-s + 9-s − 10-s + 6·13-s + 16-s − 9·17-s + 18-s − 20-s + 25-s + 6·26-s + 29-s + 32-s − 9·34-s + 36-s − 9·37-s − 40-s − 17·41-s − 45-s − 11·49-s + 50-s + 6·52-s + 2·53-s + 58-s − 8·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.66·13-s + 1/4·16-s − 2.18·17-s + 0.235·18-s − 0.223·20-s + 1/5·25-s + 1.17·26-s + 0.185·29-s + 0.176·32-s − 1.54·34-s + 1/6·36-s − 1.47·37-s − 0.158·40-s − 2.65·41-s − 0.149·45-s − 1.57·49-s + 0.141·50-s + 0.832·52-s + 0.274·53-s + 0.131·58-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(676000\)    =    \(2^{5} \cdot 5^{3} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(43.1023\)
Root analytic conductor: \(2.56227\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 676000,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
5$C_1$ \( 1 + T \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \) 2.7.a_l
11$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.11.a_b
17$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.17.j_cc
19$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.19.a_av
23$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.23.a_u
29$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ab_cg
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.j_dk
41$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.41.r_fs
43$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.43.a_ab
47$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \) 2.47.a_cn
53$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.ac_h
59$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.59.a_az
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.67.a_cc
71$C_2^2$ \( 1 + 84 T^{2} + p^{2} T^{4} \) 2.71.a_dg
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.73.o_go
79$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \) 2.79.a_adl
83$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \) 2.83.a_dk
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.ac_go
97$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.e_gr
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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

−8.309625672312158340554356809491, −7.53188973934140555194995201447, −7.08229322342209421248512414149, −6.68444671578711445880958553517, −6.35653640250017573708521597286, −5.95838679624262946438699958117, −5.17317465270278702961201943965, −4.85085614485227177991607023867, −4.30661404015640392169939513781, −3.85658690067749883024432158605, −3.36385906192802321906274894889, −2.84967285043712072283511780278, −1.86486832522701246717795624225, −1.50121331609937749932637451137, 0, 1.50121331609937749932637451137, 1.86486832522701246717795624225, 2.84967285043712072283511780278, 3.36385906192802321906274894889, 3.85658690067749883024432158605, 4.30661404015640392169939513781, 4.85085614485227177991607023867, 5.17317465270278702961201943965, 5.95838679624262946438699958117, 6.35653640250017573708521597286, 6.68444671578711445880958553517, 7.08229322342209421248512414149, 7.53188973934140555194995201447, 8.309625672312158340554356809491

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