Properties

Degree 4
Conductor $ 2^{5} \cdot 3 \cdot 7 \cdot 149 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s + 3·7-s − 8-s − 2·9-s − 10-s + 11-s + 3·13-s − 3·14-s + 16-s + 7·17-s + 2·18-s − 3·19-s + 20-s − 22-s + 3·23-s − 6·25-s − 3·26-s + 3·27-s + 3·28-s − 4·29-s + 13·31-s − 32-s − 7·34-s + 3·35-s − 2·36-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.301·11-s + 0.832·13-s − 0.801·14-s + 1/4·16-s + 1.69·17-s + 0.471·18-s − 0.688·19-s + 0.223·20-s − 0.213·22-s + 0.625·23-s − 6/5·25-s − 0.588·26-s + 0.577·27-s + 0.566·28-s − 0.742·29-s + 2.33·31-s − 0.176·32-s − 1.20·34-s + 0.507·35-s − 1/3·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100128 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100128 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100128\)    =    \(2^{5} \cdot 3 \cdot 7 \cdot 149\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100128} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 100128,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.450704010$
$L(\frac12)$  $\approx$  $1.450704010$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7,\;149\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7,\;149\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( 1 + T \)
3$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + T + p T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
149$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good5$D_{4}$ \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 7 T + 41 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T - 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 13 T + 88 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$V_4$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T + 28 T^{2} - p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
61$D_{4}$ \( 1 + 3 T + 66 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 97 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 138 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 64 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 5 T + 88 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.0826565974, −13.6138590041, −13.4434829097, −12.5821726212, −12.0671880765, −11.8435649212, −11.4385723079, −10.7866182766, −10.5609363285, −10.0970568167, −9.46115563777, −9.10882947079, −8.45574155852, −8.20885810146, −7.7846061606, −7.2195649285, −6.50478569661, −5.86464627654, −5.75295061052, −4.88157739812, −4.30159152268, −3.40950455735, −2.780771754, −1.81211203537, −1.11155344413, 1.11155344413, 1.81211203537, 2.780771754, 3.40950455735, 4.30159152268, 4.88157739812, 5.75295061052, 5.86464627654, 6.50478569661, 7.2195649285, 7.7846061606, 8.20885810146, 8.45574155852, 9.10882947079, 9.46115563777, 10.0970568167, 10.5609363285, 10.7866182766, 11.4385723079, 11.8435649212, 12.0671880765, 12.5821726212, 13.4434829097, 13.6138590041, 14.0826565974

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