Properties

Degree $4$
Conductor $656100$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 5-s + 4·7-s − 8-s + 10-s − 2·13-s + 4·14-s − 16-s + 12·17-s − 8·19-s − 2·26-s + 6·29-s − 8·31-s + 12·34-s + 4·35-s + 4·37-s − 8·38-s − 40-s + 6·41-s + 4·43-s + 7·49-s − 12·53-s − 4·56-s + 6·58-s + 10·61-s − 8·62-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.447·5-s + 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.554·13-s + 1.06·14-s − 1/4·16-s + 2.91·17-s − 1.83·19-s − 0.392·26-s + 1.11·29-s − 1.43·31-s + 2.05·34-s + 0.676·35-s + 0.657·37-s − 1.29·38-s − 0.158·40-s + 0.937·41-s + 0.609·43-s + 49-s − 1.64·53-s − 0.534·56-s + 0.787·58-s + 1.28·61-s − 1.01·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(656100\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{810} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 656100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.74518\)
\(L(\frac12)\) \(\approx\) \(3.74518\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−10.45770751965996309667976196006, −10.16869856628037836559569037510, −9.564595486171175393148115127894, −9.325404034830167114243670308657, −8.705987934118008975145398882829, −8.107344393380012462247919653855, −8.011981354759810871286767422460, −7.59350777553213198521671635243, −6.99087303284686748225062307712, −6.43946004256827292668174466088, −5.88353665570418327874869283063, −5.43089450887056725507891590063, −5.29301769457329521211234513986, −4.45189103795127144186774679254, −4.39869126149824189045523808247, −3.59005531724299466822096762104, −3.03765476391786451010987420524, −2.30541681834631813823315145256, −1.73529111796142686112091867168, −0.924757195414022668683002173340, 0.924757195414022668683002173340, 1.73529111796142686112091867168, 2.30541681834631813823315145256, 3.03765476391786451010987420524, 3.59005531724299466822096762104, 4.39869126149824189045523808247, 4.45189103795127144186774679254, 5.29301769457329521211234513986, 5.43089450887056725507891590063, 5.88353665570418327874869283063, 6.43946004256827292668174466088, 6.99087303284686748225062307712, 7.59350777553213198521671635243, 8.011981354759810871286767422460, 8.107344393380012462247919653855, 8.705987934118008975145398882829, 9.325404034830167114243670308657, 9.564595486171175393148115127894, 10.16869856628037836559569037510, 10.45770751965996309667976196006

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