Properties

Degree $4$
Conductor $6615$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 5-s + 9-s − 12-s − 15-s − 3·16-s + 4·17-s + 20-s − 2·25-s − 27-s + 36-s − 4·37-s − 12·41-s + 8·43-s + 45-s + 3·48-s − 7·49-s − 4·51-s + 8·59-s − 60-s − 7·64-s − 8·67-s + 4·68-s + 2·75-s − 3·80-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 0.447·5-s + 1/3·9-s − 0.288·12-s − 0.258·15-s − 3/4·16-s + 0.970·17-s + 0.223·20-s − 2/5·25-s − 0.192·27-s + 1/6·36-s − 0.657·37-s − 1.87·41-s + 1.21·43-s + 0.149·45-s + 0.433·48-s − 49-s − 0.560·51-s + 1.04·59-s − 0.129·60-s − 7/8·64-s − 0.977·67-s + 0.485·68-s + 0.230·75-s − 0.335·80-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6615 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6615\)    =    \(3^{3} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6615} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6615,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8899103665\)
\(L(\frac12)\) \(\approx\) \(0.8899103665\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
7$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
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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

−11.92650335429280363827949720341, −11.40400582954726890995145174500, −10.86545317284561673034688613344, −10.21657329845347327426449130856, −9.825927791305767909601870766303, −9.117135405500064951174816498766, −8.406884710420726100483844261267, −7.65001517912343532935127686469, −6.98246206054068179852838099217, −6.44841945124364554000865732178, −5.69497291084151422845642595426, −5.12456625206896000687069384338, −4.15801363831078542257449591759, −3.06014077196793113173595850965, −1.80293267241452516983333339657, 1.80293267241452516983333339657, 3.06014077196793113173595850965, 4.15801363831078542257449591759, 5.12456625206896000687069384338, 5.69497291084151422845642595426, 6.44841945124364554000865732178, 6.98246206054068179852838099217, 7.65001517912343532935127686469, 8.406884710420726100483844261267, 9.117135405500064951174816498766, 9.825927791305767909601870766303, 10.21657329845347327426449130856, 10.86545317284561673034688613344, 11.40400582954726890995145174500, 11.92650335429280363827949720341

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