Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s − 12-s − 13-s − 4·14-s + 16-s − 17-s − 18-s − 4·19-s − 4·21-s + 24-s + 26-s − 27-s + 4·28-s + 6·29-s − 4·31-s − 32-s + 34-s + 36-s − 2·37-s + 4·38-s + 39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.872·21-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-s − 0.328·37-s + 0.648·38-s + 0.160·39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 33150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(33150\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{33150} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 33150,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
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

−15.47149621193172, −14.78047849404784, −14.26797188366187, −13.92824344850509, −12.96257004262222, −12.51464347016795, −11.99928443449417, −11.32522641150379, −11.12452328058458, −10.44135553246304, −10.20346419114269, −9.145421903809773, −8.937335046592831, −8.178205784240327, −7.685925586018472, −7.292514425327239, −6.427403978428620, −6.032582162548905, −5.235162457342858, −4.636304939444582, −4.256684195401064, −3.187342938967011, −2.290113986734624, −1.722593212170246, −1.005461586667940, 0, 1.005461586667940, 1.722593212170246, 2.290113986734624, 3.187342938967011, 4.256684195401064, 4.636304939444582, 5.235162457342858, 6.032582162548905, 6.427403978428620, 7.292514425327239, 7.685925586018472, 8.178205784240327, 8.937335046592831, 9.145421903809773, 10.20346419114269, 10.44135553246304, 11.12452328058458, 11.32522641150379, 11.99928443449417, 12.51464347016795, 12.96257004262222, 13.92824344850509, 14.26797188366187, 14.78047849404784, 15.47149621193172

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