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  − 2.45·2-s + 4.04·4-s + 2.05·5-s − 7-s − 5.01·8-s − 5.04·10-s − 0.206·13-s + 2.45·14-s + 4.24·16-s − 0.813·17-s − 6.28·19-s + 8.28·20-s − 5.11·23-s − 0.793·25-s + 0.507·26-s − 4.04·28-s + 3.87·29-s + 6.49·31-s − 0.406·32-s + 2·34-s − 2.05·35-s + 1.79·37-s + 15.4·38-s − 10.2·40-s + 1.82·41-s − 10.0·43-s + 12.5·46-s + ⋯
L(s)  = 1  − 1.73·2-s + 2.02·4-s + 0.917·5-s − 0.377·7-s − 1.77·8-s − 1.59·10-s − 0.0572·13-s + 0.656·14-s + 1.06·16-s − 0.197·17-s − 1.44·19-s + 1.85·20-s − 1.06·23-s − 0.158·25-s + 0.0995·26-s − 0.763·28-s + 0.720·29-s + 1.16·31-s − 0.0719·32-s + 0.342·34-s − 0.346·35-s + 0.294·37-s + 2.50·38-s − 1.62·40-s + 0.285·41-s − 1.53·43-s + 1.85·46-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 \( 1 + 2.45T + 2T^{2} \)
5 \( 1 - 2.05T + 5T^{2} \)
13 \( 1 + 0.206T + 13T^{2} \)
17 \( 1 + 0.813T + 17T^{2} \)
19 \( 1 + 6.28T + 19T^{2} \)
23 \( 1 + 5.11T + 23T^{2} \)
29 \( 1 - 3.87T + 29T^{2} \)
31 \( 1 - 6.49T + 31T^{2} \)
37 \( 1 - 1.79T + 37T^{2} \)
41 \( 1 - 1.82T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 8.79T + 47T^{2} \)
53 \( 1 - 3.08T + 53T^{2} \)
59 \( 1 + 2.66T + 59T^{2} \)
61 \( 1 + 3.58T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 + 7.79T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 5.72T + 89T^{2} \)
97 \( 1 + 8.49T + 97T^{2} \)
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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

−7.88744943254143475652207869559, −6.75604347350592081994846740164, −6.47616294590215762609440452345, −5.85442767220682786232463519952, −4.77058062283872375263116064346, −3.76217520382456529349727985270, −2.46429841429380939934748907626, −2.17322665307442663212965261263, −1.11252404908101992539910229405, 0, 1.11252404908101992539910229405, 2.17322665307442663212965261263, 2.46429841429380939934748907626, 3.76217520382456529349727985270, 4.77058062283872375263116064346, 5.85442767220682786232463519952, 6.47616294590215762609440452345, 6.75604347350592081994846740164, 7.88744943254143475652207869559

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