Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 61 $
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-s + 3-s + 4-s + 0.871·5-s + 6-s + 1.79·7-s + 8-s + 9-s + 0.871·10-s + 11-s + 12-s + 6.05·13-s + 1.79·14-s + 0.871·15-s + 16-s + 4.28·17-s + 18-s − 0.502·19-s + 0.871·20-s + 1.79·21-s + 22-s − 0.329·23-s + 24-s − 4.24·25-s + 6.05·26-s + 27-s + 1.79·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.389·5-s + 0.408·6-s + 0.677·7-s + 0.353·8-s + 0.333·9-s + 0.275·10-s + 0.301·11-s + 0.288·12-s + 1.67·13-s + 0.478·14-s + 0.225·15-s + 0.250·16-s + 1.03·17-s + 0.235·18-s − 0.115·19-s + 0.194·20-s + 0.390·21-s + 0.213·22-s − 0.0686·23-s + 0.204·24-s − 0.848·25-s + 1.18·26-s + 0.192·27-s + 0.338·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4026\)    =    \(2 \cdot 3 \cdot 11 \cdot 61\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4026} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.063181507$
$L(\frac12)$  $\approx$  $5.063181507$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11,\;61\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;61\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 - T \)
61 \( 1 + T \)
good5 \( 1 - 0.871T + 5T^{2} \)
7 \( 1 - 1.79T + 7T^{2} \)
13 \( 1 - 6.05T + 13T^{2} \)
17 \( 1 - 4.28T + 17T^{2} \)
19 \( 1 + 0.502T + 19T^{2} \)
23 \( 1 + 0.329T + 23T^{2} \)
29 \( 1 - 0.826T + 29T^{2} \)
31 \( 1 - 2.54T + 31T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + 4.90T + 43T^{2} \)
47 \( 1 + 9.19T + 47T^{2} \)
53 \( 1 - 0.198T + 53T^{2} \)
59 \( 1 + 5.77T + 59T^{2} \)
67 \( 1 - 3.24T + 67T^{2} \)
71 \( 1 + 7.63T + 71T^{2} \)
73 \( 1 - 7.93T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 5.33T + 83T^{2} \)
89 \( 1 + 4.51T + 89T^{2} \)
97 \( 1 - 9.53T + 97T^{2} \)
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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

−8.285991155372446622166560903016, −7.87043896815039480677559110008, −6.82972007900814041715055774495, −6.17479565279631307392091837202, −5.44584322844686584390204802668, −4.64501115389542575382755736215, −3.68392690007273434306434139564, −3.23661165563173644875372829011, −1.92683784393446946269315239326, −1.32736676760791531120739594514, 1.32736676760791531120739594514, 1.92683784393446946269315239326, 3.23661165563173644875372829011, 3.68392690007273434306434139564, 4.64501115389542575382755736215, 5.44584322844686584390204802668, 6.17479565279631307392091837202, 6.82972007900814041715055774495, 7.87043896815039480677559110008, 8.285991155372446622166560903016

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