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 − 4·7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s + 4·14-s + 15-s + 16-s − 17-s − 18-s − 8·19-s − 20-s + 4·21-s − 4·22-s − 4·23-s + 24-s + 25-s − 27-s − 4·28-s + 2·29-s − 30-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 + 1.20·11-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 − 1.83·19-s − 0.223·20-s + 0.872·21-s − 0.852·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.755·28-s + 0.371·29-s − 0.182·30-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.41846572146301, −13.49006102191791, −13.04521559558885, −12.46793436803790, −12.23652685885555, −11.67889965282299, −11.15579196017285, −10.54798192271949, −10.26465579410358, −9.627579826154008, −9.229868531835670, −8.641467615833400, −8.316934291789095, −7.406977565407490, −6.954725321775128, −6.571318466225495, −6.045228910812842, −5.780382983222302, −4.630556347656741, −4.127927317439454, −3.699562105438411, −2.959909417383617, −2.243341386083223, −1.510341020874783, −0.5975075764589803, 0, 0.5975075764589803, 1.510341020874783, 2.243341386083223, 2.959909417383617, 3.699562105438411, 4.127927317439454, 4.630556347656741, 5.780382983222302, 6.045228910812842, 6.571318466225495, 6.954725321775128, 7.406977565407490, 8.316934291789095, 8.641467615833400, 9.229868531835670, 9.627579826154008, 10.26465579410358, 10.54798192271949, 11.15579196017285, 11.67889965282299, 12.23652685885555, 12.46793436803790, 13.04521559558885, 13.49006102191791, 14.41846572146301

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