Properties

Degree 4
Conductor $ 3 \cdot 7 \cdot 17 \cdot 281 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 5-s + 6-s + 3·7-s + 8-s − 2·9-s − 10-s + 11-s + 12-s − 7·13-s − 3·14-s − 15-s − 16-s − 17-s + 2·18-s + 19-s − 20-s − 3·21-s − 22-s + 23-s − 24-s + 5·25-s + 7·26-s + 2·27-s − 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.94·13-s − 0.801·14-s − 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.471·18-s + 0.229·19-s − 0.223·20-s − 0.654·21-s − 0.213·22-s + 0.208·23-s − 0.204·24-s + 25-s + 1.37·26-s + 0.384·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100317 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100317 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100317\)    =    \(3 \cdot 7 \cdot 17 \cdot 281\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100317} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 100317,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;17,\;281\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7,\;17,\;281\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
17$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
281$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 24 T + p T^{2} ) \)
good2$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$V_4$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$D_{4}$ \( 1 + 3 T + 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 9 T + 62 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$V_4$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$D_{4}$ \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 106 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 12 T + 138 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
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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

−14.2375456932, −13.839840102, −13.6659195299, −12.8052653767, −12.355864969, −12.0542149047, −11.6704739931, −11.0673716364, −10.6764548363, −10.3568710546, −9.55932640819, −9.35009207201, −8.99219500317, −8.37588283092, −7.90869890745, −7.52480342389, −6.72262861966, −6.43603642751, −5.48932593759, −5.17372454168, −4.71237641976, −4.22376667072, −2.96254118559, −2.40979070849, −1.32310402902, 0, 1.32310402902, 2.40979070849, 2.96254118559, 4.22376667072, 4.71237641976, 5.17372454168, 5.48932593759, 6.43603642751, 6.72262861966, 7.52480342389, 7.90869890745, 8.37588283092, 8.99219500317, 9.35009207201, 9.55932640819, 10.3568710546, 10.6764548363, 11.0673716364, 11.6704739931, 12.0542149047, 12.355864969, 12.8052653767, 13.6659195299, 13.839840102, 14.2375456932

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