Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.77·2-s + 1.15·4-s + 2.77·5-s − 7-s + 1.49·8-s − 4.93·10-s − 4.29·13-s + 1.77·14-s − 4.97·16-s + 2.75·17-s + 1.93·19-s + 3.22·20-s − 4.37·23-s + 2.71·25-s + 7.63·26-s − 1.15·28-s − 8.62·29-s − 0.200·31-s + 5.85·32-s − 4.89·34-s − 2.77·35-s + 1.03·37-s − 3.44·38-s + 4.14·40-s − 9.60·41-s − 4.70·43-s + 7.76·46-s + ⋯
L(s)  = 1  − 1.25·2-s + 0.579·4-s + 1.24·5-s − 0.377·7-s + 0.528·8-s − 1.56·10-s − 1.19·13-s + 0.475·14-s − 1.24·16-s + 0.668·17-s + 0.444·19-s + 0.720·20-s − 0.911·23-s + 0.542·25-s + 1.49·26-s − 0.219·28-s − 1.60·29-s − 0.0360·31-s + 1.03·32-s − 0.839·34-s − 0.469·35-s + 0.170·37-s − 0.559·38-s + 0.656·40-s − 1.50·41-s − 0.717·43-s + 1.14·46-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  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9189259658$
$L(\frac12)$  $\approx$  $0.9189259658$
$L(\frac{3}{2})$   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 \( 1 + 1.77T + 2T^{2} \)
5 \( 1 - 2.77T + 5T^{2} \)
13 \( 1 + 4.29T + 13T^{2} \)
17 \( 1 - 2.75T + 17T^{2} \)
19 \( 1 - 1.93T + 19T^{2} \)
23 \( 1 + 4.37T + 23T^{2} \)
29 \( 1 + 8.62T + 29T^{2} \)
31 \( 1 + 0.200T + 31T^{2} \)
37 \( 1 - 1.03T + 37T^{2} \)
41 \( 1 + 9.60T + 41T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 - 13.0T + 47T^{2} \)
53 \( 1 - 3.90T + 53T^{2} \)
59 \( 1 - 8.55T + 59T^{2} \)
61 \( 1 + 0.988T + 61T^{2} \)
67 \( 1 + 5.41T + 67T^{2} \)
71 \( 1 - 2.01T + 71T^{2} \)
73 \( 1 + 9.97T + 73T^{2} \)
79 \( 1 - 6.29T + 79T^{2} \)
83 \( 1 - 1.72T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−7.86861358823929780983611735133, −7.34448277761961097138420246456, −6.72811709180858555395251691065, −5.75713848043594325008752808992, −5.34513795999785343226629224695, −4.36925818392980482824283991653, −3.32238354677886329507398542444, −2.21789584861546931770932780590, −1.78019484985095762085578435712, −0.57431656039733675337272187797, 0.57431656039733675337272187797, 1.78019484985095762085578435712, 2.21789584861546931770932780590, 3.32238354677886329507398542444, 4.36925818392980482824283991653, 5.34513795999785343226629224695, 5.75713848043594325008752808992, 6.72811709180858555395251691065, 7.34448277761961097138420246456, 7.86861358823929780983611735133

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