Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
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 − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 12-s + 2·13-s + 14-s − 15-s + 16-s + 6·17-s + 18-s − 4·19-s − 20-s + 21-s − 23-s + 24-s + 25-s + 2·26-s + 27-s + 28-s + 6·29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4830\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4830} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4830,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.102490550$
$L(\frac12)$  $\approx$  $4.102490550$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−17.81290128679759, −17.01204339592390, −16.49970408807995, −15.73864873712492, −15.35359002964914, −14.61899846545044, −14.18697303227767, −13.68090795502287, −12.85461904926629, −12.28677236980731, −11.86917520450928, −10.81278887887637, −10.60865304427788, −9.607003102284166, −8.850519713948010, −8.053502603905742, −7.707841715057437, −6.808732196531968, −6.055889974969118, −5.264866185044769, −4.415594090571099, −3.769846327475227, −3.055217214893271, −2.105335603345838, −1.030343680707518, 1.030343680707518, 2.105335603345838, 3.055217214893271, 3.769846327475227, 4.415594090571099, 5.264866185044769, 6.055889974969118, 6.808732196531968, 7.707841715057437, 8.053502603905742, 8.850519713948010, 9.607003102284166, 10.60865304427788, 10.81278887887637, 11.86917520450928, 12.28677236980731, 12.85461904926629, 13.68090795502287, 14.18697303227767, 14.61899846545044, 15.35359002964914, 15.73864873712492, 16.49970408807995, 17.01204339592390, 17.81290128679759

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