Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 3.20·5-s − 6-s + 8-s + 9-s + 3.20·10-s + 11-s − 12-s + 6.40·13-s − 3.20·15-s + 16-s − 17-s + 18-s − 1.15·19-s + 3.20·20-s + 22-s + 1.95·23-s − 24-s + 5.24·25-s + 6.40·26-s − 27-s + 7.24·29-s − 3.20·30-s − 10.4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.43·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 1.01·10-s + 0.301·11-s − 0.288·12-s + 1.77·13-s − 0.826·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.264·19-s + 0.715·20-s + 0.213·22-s + 0.407·23-s − 0.204·24-s + 1.04·25-s + 1.25·26-s − 0.192·27-s + 1.34·29-s − 0.584·30-s − 1.88·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3234} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3234,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.659032941\)
\(L(\frac12)\)  \(\approx\)  \(3.659032941\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 - 3.20T + 5T^{2} \)
13 \( 1 - 6.40T + 13T^{2} \)
17 \( 1 + T + 17T^{2} \)
19 \( 1 + 1.15T + 19T^{2} \)
23 \( 1 - 1.95T + 23T^{2} \)
29 \( 1 - 7.24T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 - 5.15T + 37T^{2} \)
41 \( 1 + 8.24T + 41T^{2} \)
43 \( 1 - 5.15T + 43T^{2} \)
47 \( 1 - 6.04T + 47T^{2} \)
53 \( 1 + 6.40T + 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 + 9.70T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 - 5.24T + 71T^{2} \)
73 \( 1 - 2.09T + 73T^{2} \)
79 \( 1 + 5.60T + 79T^{2} \)
83 \( 1 + 6.55T + 83T^{2} \)
89 \( 1 - 18.4T + 89T^{2} \)
97 \( 1 - 5.49T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.869720949217232543046609762806, −7.78266293829707790369868885510, −6.67464780534178965950668735202, −6.26982016991356758216391574279, −5.72642615451475320633362944333, −4.97493418352912517494582393066, −4.06029465601077444653575257761, −3.12538746655731341828080346638, −1.96624841020152280249330656048, −1.19143127551374545663880035965, 1.19143127551374545663880035965, 1.96624841020152280249330656048, 3.12538746655731341828080346638, 4.06029465601077444653575257761, 4.97493418352912517494582393066, 5.72642615451475320633362944333, 6.26982016991356758216391574279, 6.67464780534178965950668735202, 7.78266293829707790369868885510, 8.869720949217232543046609762806

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