Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 37 \cdot 53 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 9-s − 2·10-s − 5·11-s + 2·12-s − 4·13-s + 15-s − 4·16-s + 17-s − 2·18-s − 4·19-s + 2·20-s + 10·22-s − 4·25-s + 8·26-s + 27-s − 6·29-s − 2·30-s − 2·31-s + 8·32-s − 5·33-s − 2·34-s + 2·36-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s − 1.50·11-s + 0.577·12-s − 1.10·13-s + 0.258·15-s − 16-s + 0.242·17-s − 0.471·18-s − 0.917·19-s + 0.447·20-s + 2.13·22-s − 4/5·25-s + 1.56·26-s + 0.192·27-s − 1.11·29-s − 0.365·30-s − 0.359·31-s + 1.41·32-s − 0.870·33-s − 0.342·34-s + 1/3·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100011\)    =    \(3 \cdot 17 \cdot 37 \cdot 53\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100011} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 100011,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;37,\;53\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;37,\;53\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 - T \)
37 \( 1 - T \)
53 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.15567720738645, −13.38084309277400, −13.00756931406505, −12.68830412407115, −12.01850425180635, −11.20127805262287, −10.87657494994618, −10.41918477900869, −9.857216755070655, −9.540884665889834, −9.198010912566163, −8.528288978899494, −7.887137376088910, −7.766211195880739, −7.318061418223249, −6.621673028828857, −5.954860277879920, −5.360879452933678, −4.728608546940415, −4.209634910229434, −3.316939498460537, −2.587345288808076, −2.135923291881858, −1.767745531063158, −0.6647904396289338, 0, 0.6647904396289338, 1.767745531063158, 2.135923291881858, 2.587345288808076, 3.316939498460537, 4.209634910229434, 4.728608546940415, 5.360879452933678, 5.954860277879920, 6.621673028828857, 7.318061418223249, 7.766211195880739, 7.887137376088910, 8.528288978899494, 9.198010912566163, 9.540884665889834, 9.857216755070655, 10.41918477900869, 10.87657494994618, 11.20127805262287, 12.01850425180635, 12.68830412407115, 13.00756931406505, 13.38084309277400, 14.15567720738645

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