Properties

Label 4-3087-1.1-c1e2-0-1
Degree $4$
Conductor $3087$
Sign $1$
Analytic cond. $0.196829$
Root an. cond. $0.666074$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 7-s − 3·9-s − 3·16-s + 2·25-s − 28-s − 3·36-s − 4·37-s + 8·43-s + 49-s + 3·63-s − 7·64-s − 8·67-s + 16·79-s + 9·81-s + 2·100-s − 4·109-s + 3·112-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 9·144-s − 4·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.377·7-s − 9-s − 3/4·16-s + 2/5·25-s − 0.188·28-s − 1/2·36-s − 0.657·37-s + 1.21·43-s + 1/7·49-s + 0.377·63-s − 7/8·64-s − 0.977·67-s + 1.80·79-s + 81-s + 1/5·100-s − 0.383·109-s + 0.283·112-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3/4·144-s − 0.328·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3087\)    =    \(3^{2} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(0.196829\)
Root analytic conductor: \(0.666074\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3087,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7397068074\)
\(L(\frac12)\) \(\approx\) \(0.7397068074\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + p T^{2} \)
7$C_1$ \( 1 + T \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−12.71895350827724469915215914171, −12.10442524035307922634690832050, −11.65864166326525916010083071710, −10.90160972825777864716780762811, −10.65074720861343727188897199409, −9.669715572151983783854790254476, −9.074507625868623172210132263641, −8.519119235509310100736566501089, −7.68338116225197161588634653032, −6.95297562225022579313925127204, −6.27580339173841469061935477107, −5.57679107198489813619663746991, −4.59947820001660719392658595822, −3.39358438112305592317655443101, −2.39777753602607747256751042107, 2.39777753602607747256751042107, 3.39358438112305592317655443101, 4.59947820001660719392658595822, 5.57679107198489813619663746991, 6.27580339173841469061935477107, 6.95297562225022579313925127204, 7.68338116225197161588634653032, 8.519119235509310100736566501089, 9.074507625868623172210132263641, 9.669715572151983783854790254476, 10.65074720861343727188897199409, 10.90160972825777864716780762811, 11.65864166326525916010083071710, 12.10442524035307922634690832050, 12.71895350827724469915215914171

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