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 − 3.23·5-s + 6-s + 4.29·7-s + 8-s + 9-s − 3.23·10-s + 11-s + 12-s + 2.88·13-s + 4.29·14-s − 3.23·15-s + 16-s − 1.40·17-s + 18-s + 3.85·19-s − 3.23·20-s + 4.29·21-s + 22-s + 2.00·23-s + 24-s + 5.45·25-s + 2.88·26-s + 27-s + 4.29·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.44·5-s + 0.408·6-s + 1.62·7-s + 0.353·8-s + 0.333·9-s − 1.02·10-s + 0.301·11-s + 0.288·12-s + 0.798·13-s + 1.14·14-s − 0.834·15-s + 0.250·16-s − 0.341·17-s + 0.235·18-s + 0.885·19-s − 0.722·20-s + 0.936·21-s + 0.213·22-s + 0.417·23-s + 0.204·24-s + 1.09·25-s + 0.564·26-s + 0.192·27-s + 0.811·28-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  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.894360487$
$L(\frac12)$  $\approx$  $3.894360487$
$L(\frac{3}{2})$   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 \( 1 + 3.23T + 5T^{2} \)
7 \( 1 - 4.29T + 7T^{2} \)
13 \( 1 - 2.88T + 13T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 - 3.85T + 19T^{2} \)
23 \( 1 - 2.00T + 23T^{2} \)
29 \( 1 + 1.66T + 29T^{2} \)
31 \( 1 - 0.769T + 31T^{2} \)
37 \( 1 - 6.59T + 37T^{2} \)
41 \( 1 + 7.18T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 - 0.896T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 - 8.74T + 59T^{2} \)
67 \( 1 - 6.74T + 67T^{2} \)
71 \( 1 - 3.73T + 71T^{2} \)
73 \( 1 - 0.751T + 73T^{2} \)
79 \( 1 - 0.132T + 79T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 - 2.50T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−8.270861129180040910594834652319, −7.76907053075966367277562140838, −7.17188373758616414097430871252, −6.27924542855360796254552904000, −5.03299391848282277997241427487, −4.70026144641613899486069008410, −3.76673030255661051800391767743, −3.32611109228853758071036621066, −2.04495510877173696622475967223, −1.07464930759986874719253602787, 1.07464930759986874719253602787, 2.04495510877173696622475967223, 3.32611109228853758071036621066, 3.76673030255661051800391767743, 4.70026144641613899486069008410, 5.03299391848282277997241427487, 6.27924542855360796254552904000, 7.17188373758616414097430871252, 7.76907053075966367277562140838, 8.270861129180040910594834652319

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