Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
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 − 11-s + 12-s + 2·13-s + 14-s − 15-s + 16-s + 6·17-s + 18-s − 4·19-s − 20-s + 21-s − 22-s + 24-s + 25-s + 2·26-s + 27-s + 28-s + 6·29-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 − 0.301·11-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 − 0.213·22-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 + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2310\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2310} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2310,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.578018015$
$L(\frac12)$  $\approx$  $3.578018015$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
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 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
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

−19.08096741056027, −18.84797513039154, −17.80337504674936, −17.15384979948367, −16.31617585041507, −15.72544600824249, −15.20388984231026, −14.49614882162974, −13.96439658356321, −13.38721064718905, −12.47644794389845, −12.08473625179748, −11.28646666056281, −10.38927332005888, −10.00012784391438, −8.585135712448991, −8.360214012110582, −7.446518811767572, −6.714754552571807, −5.765606064534021, −4.914214371866187, −4.080736652079647, −3.319655104552517, −2.424880332198712, −1.165700508574877, 1.165700508574877, 2.424880332198712, 3.319655104552517, 4.080736652079647, 4.914214371866187, 5.765606064534021, 6.714754552571807, 7.446518811767572, 8.360214012110582, 8.585135712448991, 10.00012784391438, 10.38927332005888, 11.28646666056281, 12.08473625179748, 12.47644794389845, 13.38721064718905, 13.96439658356321, 14.49614882162974, 15.20388984231026, 15.72544600824249, 16.31617585041507, 17.15384979948367, 17.80337504674936, 18.84797513039154, 19.08096741056027

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