Properties

Degree $2$
Conductor $70602$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 2·5-s + 6-s + 7-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s − 6·13-s + 14-s − 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s − 2·20-s + 21-s + 4·22-s + 8·23-s + 24-s − 25-s − 6·26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.218·21-s + 0.852·22-s + 1.66·23-s + 0.204·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70602 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70602 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70602\)    =    \(2 \cdot 3 \cdot 7 \cdot 41^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{70602} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 70602,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.342189629\)
\(L(\frac12)\) \(\approx\) \(4.342189629\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 - T \)
41 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.20235755143753, −13.84228704847204, −13.02433626786762, −12.72466768141295, −12.01676749562516, −11.67771890476836, −11.50235911500948, −10.63046205953682, −10.16856698546718, −9.446011170326138, −9.056686771573416, −8.504220269194146, −7.786291140575981, −7.412376575656027, −6.881930891642296, −6.596218545841352, −5.543018318353712, −4.941863069366285, −4.653322810057570, −3.962560423705013, −3.391265514578200, −2.948012423940121, −2.145454178876078, −1.496647579968130, −0.5901047702660019, 0.5901047702660019, 1.496647579968130, 2.145454178876078, 2.948012423940121, 3.391265514578200, 3.962560423705013, 4.653322810057570, 4.941863069366285, 5.543018318353712, 6.596218545841352, 6.881930891642296, 7.412376575656027, 7.786291140575981, 8.504220269194146, 9.056686771573416, 9.446011170326138, 10.16856698546718, 10.63046205953682, 11.50235911500948, 11.67771890476836, 12.01676749562516, 12.72466768141295, 13.02433626786762, 13.84228704847204, 14.20235755143753

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