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 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s + 4·7-s − 8-s + 9-s − 10-s + 12-s − 4·14-s + 15-s + 16-s + 17-s − 18-s + 4·19-s + 20-s + 4·21-s − 24-s + 25-s + 27-s + 4·28-s + 6·29-s − 30-s + 4·31-s − 32-s − 34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.872·21-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.755·28-s + 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s − 0.171·34-s + ⋯

Functional equation

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

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  =  0
Selberg data  =  $(2,\ 86190,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.015168661$
$L(\frac12)$  $\approx$  $4.015168661$
$L(\frac{3}{2})$   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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.12247615865884, −13.51927402818667, −13.07184778526645, −12.11760731317071, −11.96414497094849, −11.45356534610673, −10.76647759176999, −10.41691279032066, −9.852255710810793, −9.448789878852595, −8.719449988965583, −8.432120784717028, −7.932846979793350, −7.561039681382013, −6.797363874980675, −6.490044211443883, −5.475088863751739, −5.196590297607964, −4.588802656795866, −3.846006569084224, −3.134424503686936, −2.494546130105137, −1.908668319815791, −1.310604864499373, −0.7516159540200971, 0.7516159540200971, 1.310604864499373, 1.908668319815791, 2.494546130105137, 3.134424503686936, 3.846006569084224, 4.588802656795866, 5.196590297607964, 5.475088863751739, 6.490044211443883, 6.797363874980675, 7.561039681382013, 7.932846979793350, 8.432120784717028, 8.719449988965583, 9.448789878852595, 9.852255710810793, 10.41691279032066, 10.76647759176999, 11.45356534610673, 11.96414497094849, 12.11760731317071, 13.07184778526645, 13.51927402818667, 14.12247615865884

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