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 − 4·11-s + 12-s + 2·14-s − 15-s + 16-s − 17-s − 18-s + 2·19-s − 20-s − 2·21-s + 4·22-s − 24-s + 25-s + 27-s − 2·28-s − 6·29-s + 30-s + 10·31-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 − 1.20·11-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 + 0.852·22-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 + 1.79·31-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 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−14.10166670156647, −13.66384138818066, −13.10745757685602, −12.68273270833281, −12.31259457079096, −11.47787825377422, −11.21920174381500, −10.53601547765267, −10.08239639918734, −9.678118529779107, −9.107416024599378, −8.728605823512093, −7.940150430706362, −7.729374398779460, −7.312097504041828, −6.610360950890565, −5.975807922912304, −5.601234755445281, −4.543664628250970, −4.307147522193536, −3.357752316580737, −2.768369334961204, −2.608392252438816, −1.589107675617727, −0.7680006162745448, 0, 0.7680006162745448, 1.589107675617727, 2.608392252438816, 2.768369334961204, 3.357752316580737, 4.307147522193536, 4.543664628250970, 5.601234755445281, 5.975807922912304, 6.610360950890565, 7.312097504041828, 7.729374398779460, 7.940150430706362, 8.728605823512093, 9.107416024599378, 9.678118529779107, 10.08239639918734, 10.53601547765267, 11.21920174381500, 11.47787825377422, 12.31259457079096, 12.68273270833281, 13.10745757685602, 13.66384138818066, 14.10166670156647

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