Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 $
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 + 13-s + 14-s + 15-s + 16-s + 6·17-s + 18-s − 4·19-s + 20-s + 21-s + 24-s + 25-s + 26-s + 27-s + 28-s + 6·29-s + 30-s − 4·31-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 + 0.277·13-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 + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.182·30-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2730\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2730} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.317191569$
$L(\frac12)$  $\approx$  $4.317191569$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
good11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
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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

−19.05773154840332, −18.20162089496725, −17.47363463222563, −16.89037234431770, −16.08144977819158, −15.56831760431853, −14.73150429664804, −14.22014552906487, −13.93481185361826, −12.97081848488124, −12.50493058393333, −11.85631277652937, −10.83647154356923, −10.41256684044859, −9.551667074653084, −8.755641950115584, −8.021216762860870, −7.316814854942166, −6.434193172927746, −5.678469523379673, −4.905591159407640, −4.002575384581066, −3.189105968162988, −2.260207109270973, −1.277784273851655, 1.277784273851655, 2.260207109270973, 3.189105968162988, 4.002575384581066, 4.905591159407640, 5.678469523379673, 6.434193172927746, 7.316814854942166, 8.021216762860870, 8.755641950115584, 9.551667074653084, 10.41256684044859, 10.83647154356923, 11.85631277652937, 12.50493058393333, 12.97081848488124, 13.93481185361826, 14.22014552906487, 14.73150429664804, 15.56831760431853, 16.08144977819158, 16.89037234431770, 17.47363463222563, 18.20162089496725, 19.05773154840332

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