Properties

Degree $4$
Conductor $9834496$
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 − 14·19-s + 7·25-s + 14·27-s + 12·29-s + 10·31-s + 10·37-s + 6·47-s + 18·53-s + 28·57-s − 18·59-s − 14·75-s − 4·81-s − 24·83-s − 24·87-s − 20·93-s − 2·103-s − 22·109-s − 20·111-s + 12·113-s + 19·121-s + 127-s + 131-s + 137-s + 139-s − 12·141-s + ⋯
L(s)  = 1  − 1.15·3-s − 9-s − 3.21·19-s + 7/5·25-s + 2.69·27-s + 2.22·29-s + 1.79·31-s + 1.64·37-s + 0.875·47-s + 2.47·53-s + 3.70·57-s − 2.34·59-s − 1.61·75-s − 4/9·81-s − 2.63·83-s − 2.57·87-s − 2.07·93-s − 0.197·103-s − 2.10·109-s − 1.89·111-s + 1.12·113-s + 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.01·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9834496\)    =    \(2^{12} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3136} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9834496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.333980347\)
\(L(\frac12)\) \(\approx\) \(1.333980347\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 107 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−8.641668440583450616374179560073, −8.536428923651488807045918370325, −8.136834600189938614514045555944, −8.128745841724463020465797181739, −6.98781947117146771214434162195, −6.96433645252916185442211899555, −6.57260244251443030947066863262, −6.06830755076197169956216885626, −5.92695331126394683189090261644, −5.72989050044370266438141976182, −4.77816580377407432374688501531, −4.77543365454055639206429654275, −4.36755663456722172677625188865, −4.03709187508633558527002261586, −2.98877441346360831430960445748, −2.78938344211452363029074565718, −2.55818084243568342478986689104, −1.76310070209963987818281819017, −0.74451300334969369936724884526, −0.58600254898219169023906617950, 0.58600254898219169023906617950, 0.74451300334969369936724884526, 1.76310070209963987818281819017, 2.55818084243568342478986689104, 2.78938344211452363029074565718, 2.98877441346360831430960445748, 4.03709187508633558527002261586, 4.36755663456722172677625188865, 4.77543365454055639206429654275, 4.77816580377407432374688501531, 5.72989050044370266438141976182, 5.92695331126394683189090261644, 6.06830755076197169956216885626, 6.57260244251443030947066863262, 6.96433645252916185442211899555, 6.98781947117146771214434162195, 8.128745841724463020465797181739, 8.136834600189938614514045555944, 8.536428923651488807045918370325, 8.641668440583450616374179560073

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