Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 31^{2} $
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 − 2·5-s + 6-s + 4·7-s − 3·8-s + 9-s − 2·10-s − 11-s − 12-s + 2·13-s + 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s + 2·20-s + 4·21-s − 22-s − 8·23-s − 3·24-s − 25-s + 2·26-s + 27-s − 4·28-s + 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.447·20-s + 0.872·21-s − 0.213·22-s − 1.66·23-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 31713 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 31713 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31713\)    =    \(3 \cdot 11 \cdot 31^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{31713} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 31713,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
11 \( 1 + T \)
31 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−15.31768413868313, −14.62100611387198, −14.25452431539289, −13.74650733641698, −13.60205444089597, −12.54467396828532, −12.21777007743641, −11.82623951394482, −11.22407361538637, −10.56199313212551, −10.04402293882880, −9.273134158489500, −8.622761678558068, −8.200644395525594, −7.903054271475804, −7.331614004642958, −6.354038892182273, −5.767498140896010, −5.021964160031570, −4.620997010808923, −3.947561591804752, −3.632877866058986, −2.784747244170823, −1.948583027536287, −1.117699876178469, 0, 1.117699876178469, 1.948583027536287, 2.784747244170823, 3.632877866058986, 3.947561591804752, 4.620997010808923, 5.021964160031570, 5.767498140896010, 6.354038892182273, 7.331614004642958, 7.903054271475804, 8.200644395525594, 8.622761678558068, 9.273134158489500, 10.04402293882880, 10.56199313212551, 11.22407361538637, 11.82623951394482, 12.21777007743641, 12.54467396828532, 13.60205444089597, 13.74650733641698, 14.25452431539289, 14.62100611387198, 15.31768413868313

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