Properties

Degree $4$
Conductor $88510464$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s + 16·19-s − 8·25-s − 4·27-s − 8·31-s + 8·37-s + 8·47-s + 12·53-s − 32·57-s + 16·59-s + 16·75-s + 5·81-s + 24·83-s + 16·93-s + 8·103-s − 40·109-s − 16·111-s − 4·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s − 16·141-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 3.67·19-s − 8/5·25-s − 0.769·27-s − 1.43·31-s + 1.31·37-s + 1.16·47-s + 1.64·53-s − 4.23·57-s + 2.08·59-s + 1.84·75-s + 5/9·81-s + 2.63·83-s + 1.65·93-s + 0.788·103-s − 3.83·109-s − 1.51·111-s − 0.376·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.34·141-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(88510464\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{9408} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 88510464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.487655353\)
\(L(\frac12)\) \(\approx\) \(2.487655353\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 176 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
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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

−7.56184184779693930395337667825, −7.52459736674332421530523053279, −7.18795490131072570534455149666, −7.05102622604609218922129776780, −6.30394467882520411497166522483, −6.21033154224575324816421119453, −5.69649200753212437301240319368, −5.39742294869071738307481068308, −5.25626752390909915374149195323, −5.07718576858565706263521650520, −4.38112889045346492994828019732, −3.89276942038112794425501229498, −3.79533120852187621702184366923, −3.40364814158256812488380192464, −2.74502674537471674763041755083, −2.45566019952311633121797787484, −1.83247029596447949582387326540, −1.33133249129789716859101417493, −0.833217001646457482779875093249, −0.52551788605719588756977290166, 0.52551788605719588756977290166, 0.833217001646457482779875093249, 1.33133249129789716859101417493, 1.83247029596447949582387326540, 2.45566019952311633121797787484, 2.74502674537471674763041755083, 3.40364814158256812488380192464, 3.79533120852187621702184366923, 3.89276942038112794425501229498, 4.38112889045346492994828019732, 5.07718576858565706263521650520, 5.25626752390909915374149195323, 5.39742294869071738307481068308, 5.69649200753212437301240319368, 6.21033154224575324816421119453, 6.30394467882520411497166522483, 7.05102622604609218922129776780, 7.18795490131072570534455149666, 7.52459736674332421530523053279, 7.56184184779693930395337667825

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