Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17 $
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 − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 12-s − 2·14-s + 15-s + 16-s − 17-s − 18-s − 2·19-s − 20-s − 2·21-s − 8·23-s + 24-s + 25-s − 27-s + 2·28-s + 6·29-s − 30-s + 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.436·21-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s + 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(86190\)    =    \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{86190} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 86190,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
13 \( 1 \)
17 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.17009481272697, −13.70975781930179, −13.14764296120866, −12.35482193349957, −12.07369295102670, −11.66736972060902, −11.21096046525403, −10.62798132122817, −10.23070427662185, −9.839137948360787, −9.085691910368451, −8.526237612718664, −8.034790915985346, −7.836231586853556, −7.023121724798294, −6.459889919673686, −6.213304541470826, −5.269290204349956, −4.931768070075936, −4.198420511584625, −3.740738870121434, −2.852019322861986, −2.119866578489162, −1.570988768318831, −0.7565904053340946, 0, 0.7565904053340946, 1.570988768318831, 2.119866578489162, 2.852019322861986, 3.740738870121434, 4.198420511584625, 4.931768070075936, 5.269290204349956, 6.213304541470826, 6.459889919673686, 7.023121724798294, 7.836231586853556, 8.034790915985346, 8.526237612718664, 9.085691910368451, 9.839137948360787, 10.23070427662185, 10.62798132122817, 11.21096046525403, 11.66736972060902, 12.07369295102670, 12.35482193349957, 13.14764296120866, 13.70975781930179, 14.17009481272697

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