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 + 3.09·5-s + 6-s − 4.21·7-s + 8-s + 9-s + 3.09·10-s + 11-s + 12-s − 0.518·13-s − 4.21·14-s + 3.09·15-s + 16-s − 3.01·17-s + 18-s + 8.39·19-s + 3.09·20-s − 4.21·21-s + 22-s − 0.337·23-s + 24-s + 4.55·25-s − 0.518·26-s + 27-s − 4.21·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.38·5-s + 0.408·6-s − 1.59·7-s + 0.353·8-s + 0.333·9-s + 0.977·10-s + 0.301·11-s + 0.288·12-s − 0.143·13-s − 1.12·14-s + 0.798·15-s + 0.250·16-s − 0.730·17-s + 0.235·18-s + 1.92·19-s + 0.691·20-s − 0.919·21-s + 0.213·22-s − 0.0703·23-s + 0.204·24-s + 0.911·25-s − 0.101·26-s + 0.192·27-s − 0.796·28-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  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.446082977$
$L(\frac12)$  $\approx$  $4.446082977$
$L(\frac{3}{2})$   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 \( 1 - 3.09T + 5T^{2} \)
7 \( 1 + 4.21T + 7T^{2} \)
13 \( 1 + 0.518T + 13T^{2} \)
17 \( 1 + 3.01T + 17T^{2} \)
19 \( 1 - 8.39T + 19T^{2} \)
23 \( 1 + 0.337T + 23T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
31 \( 1 - 4.75T + 31T^{2} \)
37 \( 1 - 0.867T + 37T^{2} \)
41 \( 1 - 8.17T + 41T^{2} \)
43 \( 1 + 9.01T + 43T^{2} \)
47 \( 1 - 2.07T + 47T^{2} \)
53 \( 1 - 4.08T + 53T^{2} \)
59 \( 1 + 2.99T + 59T^{2} \)
67 \( 1 + 1.16T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 - 8.24T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + 8.21T + 83T^{2} \)
89 \( 1 - 8.31T + 89T^{2} \)
97 \( 1 + 4.53T + 97T^{2} \)
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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

−8.597599501659499016785429075618, −7.47094004992479715302610745211, −6.72523634874453392511499052199, −6.26564717803532544761952678882, −5.56575227144879787694298975973, −4.70908957299570958644085623952, −3.64742825584772285290382933898, −2.92208674387423020236062806379, −2.35341892986409573582347032842, −1.11249619646599480697799202311, 1.11249619646599480697799202311, 2.35341892986409573582347032842, 2.92208674387423020236062806379, 3.64742825584772285290382933898, 4.70908957299570958644085623952, 5.56575227144879787694298975973, 6.26564717803532544761952678882, 6.72523634874453392511499052199, 7.47094004992479715302610745211, 8.597599501659499016785429075618

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