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 + 2.15·5-s − 6-s − 1.48·7-s − 8-s + 9-s − 2.15·10-s − 11-s + 12-s + 5.53·13-s + 1.48·14-s + 2.15·15-s + 16-s + 2.33·17-s − 18-s + 3.42·19-s + 2.15·20-s − 1.48·21-s + 22-s + 3.87·23-s − 24-s − 0.335·25-s − 5.53·26-s + 27-s − 1.48·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.965·5-s − 0.408·6-s − 0.561·7-s − 0.353·8-s + 0.333·9-s − 0.682·10-s − 0.301·11-s + 0.288·12-s + 1.53·13-s + 0.397·14-s + 0.557·15-s + 0.250·16-s + 0.566·17-s − 0.235·18-s + 0.786·19-s + 0.482·20-s − 0.324·21-s + 0.213·22-s + 0.808·23-s − 0.204·24-s − 0.0671·25-s − 1.08·26-s + 0.192·27-s − 0.280·28-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  =  \((2,\ 4026,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.238436209\)
\(L(\frac12)\)  \(\approx\)  \(2.238436209\)
\(L(\frac{3}{2})\)   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
11 \( 1 + T \)
61 \( 1 + T \)
good5 \( 1 - 2.15T + 5T^{2} \)
7 \( 1 + 1.48T + 7T^{2} \)
13 \( 1 - 5.53T + 13T^{2} \)
17 \( 1 - 2.33T + 17T^{2} \)
19 \( 1 - 3.42T + 19T^{2} \)
23 \( 1 - 3.87T + 23T^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
31 \( 1 - 3.71T + 31T^{2} \)
37 \( 1 + 3.86T + 37T^{2} \)
41 \( 1 + 7.21T + 41T^{2} \)
43 \( 1 - 8.37T + 43T^{2} \)
47 \( 1 - 6.71T + 47T^{2} \)
53 \( 1 + 3.82T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
67 \( 1 + 4.45T + 67T^{2} \)
71 \( 1 + 5.39T + 71T^{2} \)
73 \( 1 - 1.17T + 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 - 4.96T + 83T^{2} \)
89 \( 1 + 4.59T + 89T^{2} \)
97 \( 1 - 3.58T + 97T^{2} \)
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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

−8.667219569144808506492042114215, −7.80534094676543432651863204510, −7.10280018194882537504357990979, −6.19403915789551880514900041388, −5.82089261927039453660390188922, −4.71473970971267244223937090462, −3.42072649944591451556600467429, −2.96144650604747442980172507402, −1.82143010174598262334040082536, −0.988883588123450762761338848117, 0.988883588123450762761338848117, 1.82143010174598262334040082536, 2.96144650604747442980172507402, 3.42072649944591451556600467429, 4.71473970971267244223937090462, 5.82089261927039453660390188922, 6.19403915789551880514900041388, 7.10280018194882537504357990979, 7.80534094676543432651863204510, 8.667219569144808506492042114215

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