Properties

Degree 2
Conductor $ 53 \cdot 151 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.58·2-s + 1.51·3-s + 4.68·4-s − 3.91·5-s − 3.92·6-s − 1.90·7-s − 6.94·8-s − 0.699·9-s + 10.1·10-s + 3.75·11-s + 7.10·12-s + 0.142·13-s + 4.93·14-s − 5.94·15-s + 8.58·16-s + 0.966·17-s + 1.80·18-s + 3.91·19-s − 18.3·20-s − 2.89·21-s − 9.71·22-s + 8.94·23-s − 10.5·24-s + 10.3·25-s − 0.368·26-s − 5.61·27-s − 8.93·28-s + ⋯
L(s)  = 1  − 1.82·2-s + 0.875·3-s + 2.34·4-s − 1.75·5-s − 1.60·6-s − 0.720·7-s − 2.45·8-s − 0.233·9-s + 3.20·10-s + 1.13·11-s + 2.05·12-s + 0.0395·13-s + 1.31·14-s − 1.53·15-s + 2.14·16-s + 0.234·17-s + 0.426·18-s + 0.897·19-s − 4.10·20-s − 0.631·21-s − 2.07·22-s + 1.86·23-s − 2.15·24-s + 2.06·25-s − 0.0723·26-s − 1.07·27-s − 1.68·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8003\)    =    \(53 \cdot 151\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8003} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8003,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.6671722745\)
\(L(\frac12)\)  \(\approx\)  \(0.6671722745\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{53,\;151\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{53,\;151\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad53 \( 1 + T \)
151 \( 1 - T \)
good2 \( 1 + 2.58T + 2T^{2} \)
3 \( 1 - 1.51T + 3T^{2} \)
5 \( 1 + 3.91T + 5T^{2} \)
7 \( 1 + 1.90T + 7T^{2} \)
11 \( 1 - 3.75T + 11T^{2} \)
13 \( 1 - 0.142T + 13T^{2} \)
17 \( 1 - 0.966T + 17T^{2} \)
19 \( 1 - 3.91T + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + 1.33T + 29T^{2} \)
31 \( 1 - 5.78T + 31T^{2} \)
37 \( 1 - 8.39T + 37T^{2} \)
41 \( 1 + 7.06T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 5.88T + 47T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 4.88T + 67T^{2} \)
71 \( 1 + 5.62T + 71T^{2} \)
73 \( 1 - 2.67T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + 4.48T + 83T^{2} \)
89 \( 1 + 4.37T + 89T^{2} \)
97 \( 1 - 7.22T + 97T^{2} \)
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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

−8.079232039695212068254118218390, −7.34586045275966860949877288580, −6.93733377542909957507617241948, −6.31778506773773578026875764938, −4.99201582909629591383158576649, −3.81078390717186082288728721870, −3.22782969500366565350270762877, −2.75610432884035328040383401344, −1.36772164421440277389621194783, −0.55408230845751597578255457093, 0.55408230845751597578255457093, 1.36772164421440277389621194783, 2.75610432884035328040383401344, 3.22782969500366565350270762877, 3.81078390717186082288728721870, 4.99201582909629591383158576649, 6.31778506773773578026875764938, 6.93733377542909957507617241948, 7.34586045275966860949877288580, 8.079232039695212068254118218390

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