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  − 0.669·2-s − 1.55·4-s − 2.14·5-s + 7-s + 2.37·8-s + 1.43·10-s + 2.52·13-s − 0.669·14-s + 1.50·16-s + 1.79·17-s − 6.72·19-s + 3.32·20-s + 3.16·23-s − 0.404·25-s − 1.69·26-s − 1.55·28-s + 0.924·29-s + 3.00·31-s − 5.76·32-s − 1.19·34-s − 2.14·35-s − 1.50·37-s + 4.50·38-s − 5.09·40-s − 5.56·41-s − 8.42·43-s − 2.11·46-s + ⋯
L(s)  = 1  − 0.473·2-s − 0.775·4-s − 0.958·5-s + 0.377·7-s + 0.840·8-s + 0.454·10-s + 0.701·13-s − 0.178·14-s + 0.377·16-s + 0.434·17-s − 1.54·19-s + 0.743·20-s + 0.659·23-s − 0.0808·25-s − 0.332·26-s − 0.293·28-s + 0.171·29-s + 0.539·31-s − 1.01·32-s − 0.205·34-s − 0.362·35-s − 0.247·37-s + 0.730·38-s − 0.806·40-s − 0.868·41-s − 1.28·43-s − 0.312·46-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 \( 1 + 0.669T + 2T^{2} \)
5 \( 1 + 2.14T + 5T^{2} \)
13 \( 1 - 2.52T + 13T^{2} \)
17 \( 1 - 1.79T + 17T^{2} \)
19 \( 1 + 6.72T + 19T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 - 0.924T + 29T^{2} \)
31 \( 1 - 3.00T + 31T^{2} \)
37 \( 1 + 1.50T + 37T^{2} \)
41 \( 1 + 5.56T + 41T^{2} \)
43 \( 1 + 8.42T + 43T^{2} \)
47 \( 1 + 4.39T + 47T^{2} \)
53 \( 1 + 0.667T + 53T^{2} \)
59 \( 1 - 0.368T + 59T^{2} \)
61 \( 1 - 5.01T + 61T^{2} \)
67 \( 1 + 0.902T + 67T^{2} \)
71 \( 1 - 14.8T + 71T^{2} \)
73 \( 1 - 8.03T + 73T^{2} \)
79 \( 1 - 4.05T + 79T^{2} \)
83 \( 1 - 4.05T + 83T^{2} \)
89 \( 1 - 8.30T + 89T^{2} \)
97 \( 1 - 8.51T + 97T^{2} \)
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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

−7.84010689994515108855630335645, −6.94440030328719393540001347737, −6.27858931971005371973686986573, −5.19254962837561543726553012951, −4.69372230113894755514765800926, −3.88089933079657475135014760430, −3.41098313774011483098330192541, −2.04678160554319797401908399048, −1.02370960388510578666082169785, 0, 1.02370960388510578666082169785, 2.04678160554319797401908399048, 3.41098313774011483098330192541, 3.88089933079657475135014760430, 4.69372230113894755514765800926, 5.19254962837561543726553012951, 6.27858931971005371973686986573, 6.94440030328719393540001347737, 7.84010689994515108855630335645

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