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  + 1.64·2-s + 3-s + 0.703·4-s + 3.28·5-s + 1.64·6-s − 2.13·8-s + 9-s + 5.39·10-s − 3.35·11-s + 0.703·12-s + 5.75·13-s + 3.28·15-s − 4.91·16-s + 6.12·17-s + 1.64·18-s + 3.14·19-s + 2.30·20-s − 5.52·22-s − 9.17·23-s − 2.13·24-s + 5.77·25-s + 9.46·26-s + 27-s + 0.986·29-s + 5.39·30-s − 4.50·31-s − 3.81·32-s + ⋯
L(s)  = 1  + 1.16·2-s + 0.577·3-s + 0.351·4-s + 1.46·5-s + 0.671·6-s − 0.753·8-s + 0.333·9-s + 1.70·10-s − 1.01·11-s + 0.203·12-s + 1.59·13-s + 0.847·15-s − 1.22·16-s + 1.48·17-s + 0.387·18-s + 0.720·19-s + 0.516·20-s − 1.17·22-s − 1.91·23-s − 0.435·24-s + 1.15·25-s + 1.85·26-s + 0.192·27-s + 0.183·29-s + 0.985·30-s − 0.808·31-s − 0.673·32-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  =  $(2,\ 6027,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.953693933$
$L(\frac12)$  $\approx$  $5.953693933$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good2 \( 1 - 1.64T + 2T^{2} \)
5 \( 1 - 3.28T + 5T^{2} \)
11 \( 1 + 3.35T + 11T^{2} \)
13 \( 1 - 5.75T + 13T^{2} \)
17 \( 1 - 6.12T + 17T^{2} \)
19 \( 1 - 3.14T + 19T^{2} \)
23 \( 1 + 9.17T + 23T^{2} \)
29 \( 1 - 0.986T + 29T^{2} \)
31 \( 1 + 4.50T + 31T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
43 \( 1 - 7.67T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 2.61T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 - 1.80T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + 5.92T + 71T^{2} \)
73 \( 1 + 9.66T + 73T^{2} \)
79 \( 1 + 8.44T + 79T^{2} \)
83 \( 1 - 5.92T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−8.025323616299624433510768745741, −7.36551620968759150021529444917, −6.06249860139614085810299852224, −5.89598842394088011258601660925, −5.43754807907829032527727575244, −4.35947814617686247353552591795, −3.66255108349273012079842136790, −2.87930595259475218377176235100, −2.19998535241385337635285298755, −1.12102967039295493655511340729, 1.12102967039295493655511340729, 2.19998535241385337635285298755, 2.87930595259475218377176235100, 3.66255108349273012079842136790, 4.35947814617686247353552591795, 5.43754807907829032527727575244, 5.89598842394088011258601660925, 6.06249860139614085810299852224, 7.36551620968759150021529444917, 8.025323616299624433510768745741

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