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  + 2.05·2-s + 2.23·4-s + 3.05·5-s − 7-s + 0.492·8-s + 6.29·10-s + 3.32·13-s − 2.05·14-s − 3.46·16-s − 5.37·17-s + 8.29·19-s + 6.84·20-s + 1.74·23-s + 4.35·25-s + 6.84·26-s − 2.23·28-s + 8.16·29-s − 5.44·31-s − 8.11·32-s − 11.0·34-s − 3.05·35-s + 4.53·37-s + 17.0·38-s + 1.50·40-s + 6.84·41-s − 1.84·43-s + 3.58·46-s + ⋯
L(s)  = 1  + 1.45·2-s + 1.11·4-s + 1.36·5-s − 0.377·7-s + 0.174·8-s + 1.99·10-s + 0.922·13-s − 0.550·14-s − 0.866·16-s − 1.30·17-s + 1.90·19-s + 1.53·20-s + 0.363·23-s + 0.871·25-s + 1.34·26-s − 0.423·28-s + 1.51·29-s − 0.978·31-s − 1.43·32-s − 1.89·34-s − 0.517·35-s + 0.745·37-s + 2.77·38-s + 0.238·40-s + 1.06·41-s − 0.282·43-s + 0.528·46-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  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.189693136$
$L(\frac12)$  $\approx$  $6.189693136$
$L(\frac{3}{2})$   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 \( 1 - 2.05T + 2T^{2} \)
5 \( 1 - 3.05T + 5T^{2} \)
13 \( 1 - 3.32T + 13T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
19 \( 1 - 8.29T + 19T^{2} \)
23 \( 1 - 1.74T + 23T^{2} \)
29 \( 1 - 8.16T + 29T^{2} \)
31 \( 1 + 5.44T + 31T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 - 6.84T + 41T^{2} \)
43 \( 1 + 1.84T + 43T^{2} \)
47 \( 1 - 7.28T + 47T^{2} \)
53 \( 1 - 0.985T + 53T^{2} \)
59 \( 1 + 4.52T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 - 0.170T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 - 1.62T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 3.44T + 97T^{2} \)
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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

−7.60898128925806734851700800257, −6.71904741773451816215475437153, −6.30101743454153297196635660105, −5.70229463829404540165616980856, −5.14703909004477386609851093633, −4.41506514832853475501553889364, −3.54988574335104572415166593160, −2.83414605232112325262456236295, −2.16276951777660565528606488146, −1.03969336350816771079883236663, 1.03969336350816771079883236663, 2.16276951777660565528606488146, 2.83414605232112325262456236295, 3.54988574335104572415166593160, 4.41506514832853475501553889364, 5.14703909004477386609851093633, 5.70229463829404540165616980856, 6.30101743454153297196635660105, 6.71904741773451816215475437153, 7.60898128925806734851700800257

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