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 − 0.600·5-s + 6-s + 0.748·7-s + 8-s + 9-s − 0.600·10-s + 11-s + 12-s − 3.37·13-s + 0.748·14-s − 0.600·15-s + 16-s + 1.61·17-s + 18-s + 1.26·19-s − 0.600·20-s + 0.748·21-s + 22-s + 5.33·23-s + 24-s − 4.63·25-s − 3.37·26-s + 27-s + 0.748·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.268·5-s + 0.408·6-s + 0.282·7-s + 0.353·8-s + 0.333·9-s − 0.189·10-s + 0.301·11-s + 0.288·12-s − 0.935·13-s + 0.199·14-s − 0.155·15-s + 0.250·16-s + 0.392·17-s + 0.235·18-s + 0.289·19-s − 0.134·20-s + 0.163·21-s + 0.213·22-s + 1.11·23-s + 0.204·24-s − 0.927·25-s − 0.661·26-s + 0.192·27-s + 0.141·28-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  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.919915998$
$L(\frac12)$  $\approx$  $3.919915998$
$L(\frac{3}{2})$   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 \( 1 + 0.600T + 5T^{2} \)
7 \( 1 - 0.748T + 7T^{2} \)
13 \( 1 + 3.37T + 13T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 - 5.33T + 23T^{2} \)
29 \( 1 - 3.93T + 29T^{2} \)
31 \( 1 - 6.73T + 31T^{2} \)
37 \( 1 - 0.919T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 0.371T + 43T^{2} \)
47 \( 1 - 1.78T + 47T^{2} \)
53 \( 1 + 8.73T + 53T^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
67 \( 1 - 6.96T + 67T^{2} \)
71 \( 1 - 7.20T + 71T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 - 15.8T + 79T^{2} \)
83 \( 1 - 5.44T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−8.070833752057286891239639924251, −7.86817487864206581217362484327, −6.93682435621422280690934342027, −6.30265873975693758915474329475, −5.23229366443099064572943681086, −4.66239516507539799319949507883, −3.85361825472067135642379591180, −2.99323337233510096108083679797, −2.26894028276143987109713952680, −1.03890674944333899940515016160, 1.03890674944333899940515016160, 2.26894028276143987109713952680, 2.99323337233510096108083679797, 3.85361825472067135642379591180, 4.66239516507539799319949507883, 5.23229366443099064572943681086, 6.30265873975693758915474329475, 6.93682435621422280690934342027, 7.86817487864206581217362484327, 8.070833752057286891239639924251

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