Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.35·2-s + 3.56·4-s + 3.48·5-s + 7-s + 3.69·8-s + 8.21·10-s + 3.61·11-s − 1.92·13-s + 2.35·14-s + 1.58·16-s + 6.15·17-s − 4.23·19-s + 12.4·20-s + 8.53·22-s + 7.99·23-s + 7.13·25-s − 4.55·26-s + 3.56·28-s − 2.42·29-s − 1.78·31-s − 3.64·32-s + 14.5·34-s + 3.48·35-s + 5.76·37-s − 9.99·38-s + 12.8·40-s − 6.30·41-s + ⋯
L(s)  = 1  + 1.66·2-s + 1.78·4-s + 1.55·5-s + 0.377·7-s + 1.30·8-s + 2.59·10-s + 1.09·11-s − 0.535·13-s + 0.630·14-s + 0.397·16-s + 1.49·17-s − 0.971·19-s + 2.77·20-s + 1.81·22-s + 1.66·23-s + 1.42·25-s − 0.892·26-s + 0.674·28-s − 0.450·29-s − 0.320·31-s − 0.644·32-s + 2.48·34-s + 0.588·35-s + 0.947·37-s − 1.62·38-s + 2.03·40-s − 0.985·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8001} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $9.025158544$
$L(\frac12)$  $\approx$  $9.025158544$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
127 \( 1 - T \)
good2 \( 1 - 2.35T + 2T^{2} \)
5 \( 1 - 3.48T + 5T^{2} \)
11 \( 1 - 3.61T + 11T^{2} \)
13 \( 1 + 1.92T + 13T^{2} \)
17 \( 1 - 6.15T + 17T^{2} \)
19 \( 1 + 4.23T + 19T^{2} \)
23 \( 1 - 7.99T + 23T^{2} \)
29 \( 1 + 2.42T + 29T^{2} \)
31 \( 1 + 1.78T + 31T^{2} \)
37 \( 1 - 5.76T + 37T^{2} \)
41 \( 1 + 6.30T + 41T^{2} \)
43 \( 1 + 2.65T + 43T^{2} \)
47 \( 1 + 3.09T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 + 7.38T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 8.46T + 79T^{2} \)
83 \( 1 + 8.32T + 83T^{2} \)
89 \( 1 + 5.58T + 89T^{2} \)
97 \( 1 - 3.96T + 97T^{2} \)
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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

−7.46283335747310322219204833839, −6.76290348864831564044050674846, −6.22633554636220280044741809419, −5.66142853579046542042392069735, −5.02695451710210691836185806115, −4.52548476365320712173627195693, −3.49338973596106878346617008929, −2.88758442193579161368624614648, −1.95921661274723415617720879960, −1.33564518640688250770021025030, 1.33564518640688250770021025030, 1.95921661274723415617720879960, 2.88758442193579161368624614648, 3.49338973596106878346617008929, 4.52548476365320712173627195693, 5.02695451710210691836185806115, 5.66142853579046542042392069735, 6.22633554636220280044741809419, 6.76290348864831564044050674846, 7.46283335747310322219204833839

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