Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s + 4.85·4-s − 5-s − 7-s − 7.47·8-s + 2.61·10-s − 3.23·13-s + 2.61·14-s + 9.85·16-s − 8.09·17-s + 6.23·19-s − 4.85·20-s + 6.09·23-s − 4·25-s + 8.47·26-s − 4.85·28-s − 2.38·29-s + 0.236·31-s − 10.8·32-s + 21.1·34-s + 35-s − 2.47·37-s − 16.3·38-s + 7.47·40-s + 11.1·41-s − 7.56·43-s − 15.9·46-s + ⋯
L(s)  = 1  − 1.85·2-s + 2.42·4-s − 0.447·5-s − 0.377·7-s − 2.64·8-s + 0.827·10-s − 0.897·13-s + 0.699·14-s + 2.46·16-s − 1.96·17-s + 1.43·19-s − 1.08·20-s + 1.26·23-s − 0.800·25-s + 1.66·26-s − 0.917·28-s − 0.442·29-s + 0.0423·31-s − 1.91·32-s + 3.63·34-s + 0.169·35-s − 0.406·37-s − 2.64·38-s + 1.18·40-s + 1.74·41-s − 1.15·43-s − 2.35·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 2.61T + 2T^{2} \)
5 \( 1 + T + 5T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 + 8.09T + 17T^{2} \)
19 \( 1 - 6.23T + 19T^{2} \)
23 \( 1 - 6.09T + 23T^{2} \)
29 \( 1 + 2.38T + 29T^{2} \)
31 \( 1 - 0.236T + 31T^{2} \)
37 \( 1 + 2.47T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 7.56T + 43T^{2} \)
47 \( 1 - 4.38T + 47T^{2} \)
53 \( 1 - 4.61T + 53T^{2} \)
59 \( 1 - 0.0901T + 59T^{2} \)
61 \( 1 - 5.38T + 61T^{2} \)
67 \( 1 - 7.32T + 67T^{2} \)
71 \( 1 - 4.90T + 71T^{2} \)
73 \( 1 - 9.76T + 73T^{2} \)
79 \( 1 + 8.61T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + 0.145T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−7.48085905236345556870915698685, −7.17704354450408090541549292219, −6.60010221302847682970177704701, −5.68268472682474777385594792950, −4.75039439926802169622314186159, −3.66468988033671951088633000857, −2.70026555537369215000132438275, −2.10319146656435429834705392838, −0.911068768582631026107892254663, 0, 0.911068768582631026107892254663, 2.10319146656435429834705392838, 2.70026555537369215000132438275, 3.66468988033671951088633000857, 4.75039439926802169622314186159, 5.68268472682474777385594792950, 6.60010221302847682970177704701, 7.17704354450408090541549292219, 7.48085905236345556870915698685

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