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 + 2·17-s + 18-s + 4·19-s − 20-s + 21-s − 22-s − 4·23-s − 24-s + 25-s + 2·26-s − 27-s − 28-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 + 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-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  =  $(2,\ 2310,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.037343752$
$L(\frac12)$  $\approx$  $2.037343752$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−19.05289517522378, −18.55340119321469, −17.85615793556819, −17.05744318456604, −16.32982495562451, −15.89724952206621, −15.41561450960532, −14.50492696038298, −13.87935953606847, −13.19157454178722, −12.41005695517353, −12.08707642965589, −11.10669962953342, −10.81980337707420, −9.797215508547846, −9.135194326173858, −7.840347589172237, −7.516453567510014, −6.449812519606532, −5.810082721014041, −5.133469819672676, −4.079648724669524, −3.491805983573957, −2.322069255035379, −0.8632399565020954, 0.8632399565020954, 2.322069255035379, 3.491805983573957, 4.079648724669524, 5.133469819672676, 5.810082721014041, 6.449812519606532, 7.516453567510014, 7.840347589172237, 9.135194326173858, 9.797215508547846, 10.81980337707420, 11.10669962953342, 12.08707642965589, 12.41005695517353, 13.19157454178722, 13.87935953606847, 14.50492696038298, 15.41561450960532, 15.89724952206621, 16.32982495562451, 17.05744318456604, 17.85615793556819, 18.55340119321469, 19.05289517522378

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