Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 13 \cdot 97 $
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 + 7-s + 8-s + 9-s + 10-s + 5·11-s + 12-s − 13-s + 14-s + 15-s + 16-s − 3·17-s + 18-s − 8·19-s + 20-s + 21-s + 5·22-s − 4·23-s + 24-s + 25-s − 26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s + 0.218·21-s + 1.06·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 37830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 37830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(37830\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 97\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{37830} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 37830,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.215091397$
$L(\frac12)$  $\approx$  $6.215091397$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;13,\;97\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;97\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 + T \)
97 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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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

−14.66794330537699, −14.43767552699941, −13.92179623245040, −13.24982533098280, −12.95446755374710, −12.29645983709919, −11.83264957597524, −11.19486838059708, −10.76483098221908, −10.07557245744635, −9.520589623162203, −8.927867017961444, −8.505038824407182, −7.854257443388859, −7.163161760448861, −6.459034602056771, −6.299866850776167, −5.517647796659679, −4.656862106644139, −4.118439649779603, −3.914676257423867, −2.846952768575045, −2.162675950623610, −1.809540034683127, −0.7794062114562985, 0.7794062114562985, 1.809540034683127, 2.162675950623610, 2.846952768575045, 3.914676257423867, 4.118439649779603, 4.656862106644139, 5.517647796659679, 6.299866850776167, 6.459034602056771, 7.163161760448861, 7.854257443388859, 8.505038824407182, 8.927867017961444, 9.520589623162203, 10.07557245744635, 10.76483098221908, 11.19486838059708, 11.83264957597524, 12.29645983709919, 12.95446755374710, 13.24982533098280, 13.92179623245040, 14.43767552699941, 14.66794330537699

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