Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \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 + 4-s + 5-s + 7-s + 8-s + 10-s + 11-s + 2·13-s + 14-s + 16-s + 6·17-s − 4·19-s + 20-s + 22-s + 25-s + 2·26-s + 28-s + 6·29-s − 4·31-s + 32-s + 6·34-s + 35-s + 2·37-s − 4·38-s + 40-s + 6·41-s − 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.392·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6930} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6930,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.288916287$
$L(\frac12)$  $\approx$  $4.288916287$
$L(\frac{3}{2})$   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 \( 1 \)
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 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−16.87355546052363, −16.77960634507901, −15.98238219574462, −15.33970964364609, −14.61333898912451, −14.33720630334724, −13.72851719928255, −13.01531456480835, −12.55191885292462, −11.85284260984647, −11.34046178203736, −10.48707161371258, −10.19060528368969, −9.239418020823513, −8.586388342631418, −7.862629758333126, −7.190977855425709, −6.312657489275396, −5.887829027304360, −5.126857712602204, −4.390651468913151, −3.630557377996476, −2.831667027196674, −1.879852085271210, −1.014496881447660, 1.014496881447660, 1.879852085271210, 2.831667027196674, 3.630557377996476, 4.390651468913151, 5.126857712602204, 5.887829027304360, 6.312657489275396, 7.190977855425709, 7.862629758333126, 8.586388342631418, 9.239418020823513, 10.19060528368969, 10.48707161371258, 11.34046178203736, 11.85284260984647, 12.55191885292462, 13.01531456480835, 13.72851719928255, 14.33720630334724, 14.61333898912451, 15.33970964364609, 15.98238219574462, 16.77960634507901, 16.87355546052363

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