Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{6} \cdot 5^{4} \cdot 7^{2} $
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  − 2·2-s + 3·4-s + 2·7-s − 4·8-s + 3·11-s − 2·13-s − 4·14-s + 5·16-s + 17-s + 7·19-s − 6·22-s − 3·23-s + 4·26-s + 6·28-s + 3·29-s + 7·31-s − 6·32-s − 2·34-s − 2·37-s − 14·38-s + 11·41-s + 14·43-s + 9·44-s + 6·46-s − 11·47-s + 3·49-s − 6·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s + 0.904·11-s − 0.554·13-s − 1.06·14-s + 5/4·16-s + 0.242·17-s + 1.60·19-s − 1.27·22-s − 0.625·23-s + 0.784·26-s + 1.13·28-s + 0.557·29-s + 1.25·31-s − 1.06·32-s − 0.342·34-s − 0.328·37-s − 2.27·38-s + 1.71·41-s + 2.13·43-s + 1.35·44-s + 0.884·46-s − 1.60·47-s + 3/7·49-s − 0.832·52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(89302500\)    =    \(2^{2} \cdot 3^{6} \cdot 5^{4} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{9450} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 89302500,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $2.080946498$
$L(\frac12)$  $\approx$  $2.080946498$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$D_{4}$ \( 1 - 3 T - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 36 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 50 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T - 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 11 T + 88 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 11 T + 100 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 - 5 T + 100 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - T + 118 T^{2} - p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 5 T + 128 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 25 T + 298 T^{2} + 25 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−7.75945440961521878635015619901, −7.55328977730697777566663347456, −7.30747627034380874196053803739, −7.19170628569305787856895060116, −6.42508496443011791599354783734, −6.37874294761802079250141730761, −5.84179782729380627942000723841, −5.74662113277053634053109974186, −5.07019661601460422570935549524, −4.82157164955359703962345612338, −4.39670973595914339416190351526, −4.03948931634693608489759938734, −3.42169317763331040942022624053, −3.20299263668525711166953292924, −2.53580149543667072576251740225, −2.42808887357122607320666976207, −1.77375558149517845010258689688, −1.28035722352257182438114929080, −1.01260825345890657038299204786, −0.48814950526400789213457592474, 0.48814950526400789213457592474, 1.01260825345890657038299204786, 1.28035722352257182438114929080, 1.77375558149517845010258689688, 2.42808887357122607320666976207, 2.53580149543667072576251740225, 3.20299263668525711166953292924, 3.42169317763331040942022624053, 4.03948931634693608489759938734, 4.39670973595914339416190351526, 4.82157164955359703962345612338, 5.07019661601460422570935549524, 5.74662113277053634053109974186, 5.84179782729380627942000723841, 6.37874294761802079250141730761, 6.42508496443011791599354783734, 7.19170628569305787856895060116, 7.30747627034380874196053803739, 7.55328977730697777566663347456, 7.75945440961521878635015619901

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