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 + 4·5-s − 6-s − 8-s + 9-s − 4·10-s + 11-s + 12-s + 2·13-s + 4·15-s + 16-s + 2·17-s − 18-s + 4·19-s + 4·20-s − 22-s + 8·23-s − 24-s + 11·25-s − 2·26-s + 27-s + 2·29-s − 4·30-s − 2·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.894·20-s − 0.213·22-s + 1.66·23-s − 0.204·24-s + 11/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.730·30-s − 0.359·31-s − 0.176·32-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  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.863115931$
$L(\frac12)$  $\approx$  $2.863115931$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.608718948017794829733912841514, −7.908066673185964494948274268454, −6.74333404085984723786614245803, −6.60589808110925983728983232275, −5.46172575196020262329273859280, −4.98947488697477787376362606595, −3.41920990896705159398090407919, −2.83750008917948826612427422392, −1.73178686189749874158490306737, −1.20452717923265751181000511481, 1.20452717923265751181000511481, 1.73178686189749874158490306737, 2.83750008917948826612427422392, 3.41920990896705159398090407919, 4.98947488697477787376362606595, 5.46172575196020262329273859280, 6.60589808110925983728983232275, 6.74333404085984723786614245803, 7.908066673185964494948274268454, 8.608718948017794829733912841514

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