Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 127 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.840·2-s + 3-s − 1.29·4-s + 2.74·5-s + 0.840·6-s + 7-s − 2.76·8-s + 9-s + 2.30·10-s − 5.62·11-s − 1.29·12-s − 3.65·13-s + 0.840·14-s + 2.74·15-s + 0.257·16-s − 5.86·17-s + 0.840·18-s − 2.05·19-s − 3.55·20-s + 21-s − 4.72·22-s + 2.78·23-s − 2.76·24-s + 2.54·25-s − 3.07·26-s + 27-s − 1.29·28-s + ⋯
L(s)  = 1  + 0.594·2-s + 0.577·3-s − 0.646·4-s + 1.22·5-s + 0.343·6-s + 0.377·7-s − 0.978·8-s + 0.333·9-s + 0.730·10-s − 1.69·11-s − 0.373·12-s − 1.01·13-s + 0.224·14-s + 0.709·15-s + 0.0644·16-s − 1.42·17-s + 0.198·18-s − 0.472·19-s − 0.794·20-s + 0.218·21-s − 1.00·22-s + 0.580·23-s − 0.565·24-s + 0.509·25-s − 0.602·26-s + 0.192·27-s − 0.244·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2667\)    =    \(3 \cdot 7 \cdot 127\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2667} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 2667,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;127\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;127\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
127 \( 1 + T \)
good2 \( 1 - 0.840T + 2T^{2} \)
5 \( 1 - 2.74T + 5T^{2} \)
11 \( 1 + 5.62T + 11T^{2} \)
13 \( 1 + 3.65T + 13T^{2} \)
17 \( 1 + 5.86T + 17T^{2} \)
19 \( 1 + 2.05T + 19T^{2} \)
23 \( 1 - 2.78T + 23T^{2} \)
29 \( 1 + 4.83T + 29T^{2} \)
31 \( 1 + 6.87T + 31T^{2} \)
37 \( 1 + 8.71T + 37T^{2} \)
41 \( 1 + 4.09T + 41T^{2} \)
43 \( 1 + 2.48T + 43T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 - 4.92T + 53T^{2} \)
59 \( 1 + 9.41T + 59T^{2} \)
61 \( 1 - 4.96T + 61T^{2} \)
67 \( 1 - 4.57T + 67T^{2} \)
71 \( 1 - 2.13T + 71T^{2} \)
73 \( 1 - 9.59T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + 5.06T + 83T^{2} \)
89 \( 1 - 2.73T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−8.796881824827830843365819778475, −7.70238267445897695309459818605, −6.96896386281210364013217363053, −5.85658754025261944095891012238, −5.19112249410786507410836165283, −4.75023222838737545033259919734, −3.63321525868860762612857970041, −2.50431225623514826276977388219, −2.04887033637200668197075951327, 0, 2.04887033637200668197075951327, 2.50431225623514826276977388219, 3.63321525868860762612857970041, 4.75023222838737545033259919734, 5.19112249410786507410836165283, 5.85658754025261944095891012238, 6.96896386281210364013217363053, 7.70238267445897695309459818605, 8.796881824827830843365819778475

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