Properties

Degree $4$
Conductor $5531904$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 2·11-s − 8·13-s − 2·15-s − 6·17-s + 8·19-s − 6·23-s + 5·25-s − 27-s − 20·29-s + 4·31-s + 2·33-s − 6·37-s − 8·39-s − 12·41-s − 8·43-s + 8·47-s − 6·51-s − 2·53-s − 4·55-s + 8·57-s − 4·59-s + 8·61-s + 16·65-s − 8·67-s − 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.603·11-s − 2.21·13-s − 0.516·15-s − 1.45·17-s + 1.83·19-s − 1.25·23-s + 25-s − 0.192·27-s − 3.71·29-s + 0.718·31-s + 0.348·33-s − 0.986·37-s − 1.28·39-s − 1.87·41-s − 1.21·43-s + 1.16·47-s − 0.840·51-s − 0.274·53-s − 0.539·55-s + 1.05·57-s − 0.520·59-s + 1.02·61-s + 1.98·65-s − 0.977·67-s − 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 6 T - T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 2 T - 49 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 4 T - 57 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
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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

−8.952092102264478939417605183636, −8.347745660567478483024884182469, −7.900395278303626442563058877986, −7.73274766930444364787518803716, −7.18256948249878411898283374452, −7.07452671216611716348857875793, −6.72431846429709112841254143167, −6.09520942031107164084600426778, −5.44037996290502931606410282014, −5.18439119880678360445105408767, −4.87541017799406411303651681621, −4.26700606129793569587102570752, −3.71123640615818591537225910282, −3.65366053361810572148739638283, −2.98835528622851507097297977176, −2.39551089807479565871362914581, −2.01201944768766425712037358620, −1.40825066593104697181540410898, 0, 0, 1.40825066593104697181540410898, 2.01201944768766425712037358620, 2.39551089807479565871362914581, 2.98835528622851507097297977176, 3.65366053361810572148739638283, 3.71123640615818591537225910282, 4.26700606129793569587102570752, 4.87541017799406411303651681621, 5.18439119880678360445105408767, 5.44037996290502931606410282014, 6.09520942031107164084600426778, 6.72431846429709112841254143167, 7.07452671216611716348857875793, 7.18256948249878411898283374452, 7.73274766930444364787518803716, 7.900395278303626442563058877986, 8.347745660567478483024884182469, 8.952092102264478939417605183636

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