Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.78·2-s − 3-s + 5.73·4-s − 0.739·5-s − 2.78·6-s + 10.3·8-s + 9-s − 2.05·10-s + 4.44·11-s − 5.73·12-s − 0.996·13-s + 0.739·15-s + 17.4·16-s + 1.26·17-s + 2.78·18-s + 1.98·19-s − 4.24·20-s + 12.3·22-s + 0.818·23-s − 10.3·24-s − 4.45·25-s − 2.77·26-s − 27-s − 0.261·29-s + 2.05·30-s + 7.23·31-s + 27.6·32-s + ⋯
L(s)  = 1  + 1.96·2-s − 0.577·3-s + 2.86·4-s − 0.330·5-s − 1.13·6-s + 3.67·8-s + 0.333·9-s − 0.650·10-s + 1.33·11-s − 1.65·12-s − 0.276·13-s + 0.190·15-s + 4.35·16-s + 0.306·17-s + 0.655·18-s + 0.454·19-s − 0.948·20-s + 2.63·22-s + 0.170·23-s − 2.12·24-s − 0.890·25-s − 0.543·26-s − 0.192·27-s − 0.0485·29-s + 0.375·30-s + 1.29·31-s + 4.89·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6027} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.068978950$
$L(\frac12)$  $\approx$  $7.068978950$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 2.78T + 2T^{2} \)
5 \( 1 + 0.739T + 5T^{2} \)
11 \( 1 - 4.44T + 11T^{2} \)
13 \( 1 + 0.996T + 13T^{2} \)
17 \( 1 - 1.26T + 17T^{2} \)
19 \( 1 - 1.98T + 19T^{2} \)
23 \( 1 - 0.818T + 23T^{2} \)
29 \( 1 + 0.261T + 29T^{2} \)
31 \( 1 - 7.23T + 31T^{2} \)
37 \( 1 + 7.70T + 37T^{2} \)
43 \( 1 + 4.72T + 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 7.87T + 59T^{2} \)
61 \( 1 + 6.85T + 61T^{2} \)
67 \( 1 - 4.70T + 67T^{2} \)
71 \( 1 + 5.60T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 1.81T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 - 7.76T + 97T^{2} \)
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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

−7.53838567895261593153834418784, −7.12884796453041388544886857534, −6.30539489787543668420464389942, −5.95492960170484108456202274905, −5.04033407066057212236512969474, −4.52535159977197815661935834043, −3.76438383876618521482713210248, −3.22056578746422423050654406145, −2.08589634681692334158065167221, −1.16924115965295672259605224900, 1.16924115965295672259605224900, 2.08589634681692334158065167221, 3.22056578746422423050654406145, 3.76438383876618521482713210248, 4.52535159977197815661935834043, 5.04033407066057212236512969474, 5.95492960170484108456202274905, 6.30539489787543668420464389942, 7.12884796453041388544886857534, 7.53838567895261593153834418784

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