Properties

Label 2-402-1.1-c1-0-7
Degree $2$
Conductor $402$
Sign $1$
Analytic cond. $3.20998$
Root an. cond. $1.79164$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 0.710·5-s + 6-s − 1.81·7-s + 8-s + 9-s + 0.710·10-s + 0.578·11-s + 12-s + 4.72·13-s − 1.81·14-s + 0.710·15-s + 16-s + 4.20·17-s + 18-s − 2·19-s + 0.710·20-s − 1.81·21-s + 0.578·22-s − 5.44·23-s + 24-s − 4.49·25-s + 4.72·26-s + 27-s − 1.81·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.317·5-s + 0.408·6-s − 0.685·7-s + 0.353·8-s + 0.333·9-s + 0.224·10-s + 0.174·11-s + 0.288·12-s + 1.31·13-s − 0.484·14-s + 0.183·15-s + 0.250·16-s + 1.01·17-s + 0.235·18-s − 0.458·19-s + 0.158·20-s − 0.395·21-s + 0.123·22-s − 1.13·23-s + 0.204·24-s − 0.898·25-s + 0.927·26-s + 0.192·27-s − 0.342·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 402 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(402\)    =    \(2 \cdot 3 \cdot 67\)
Sign: $1$
Analytic conductor: \(3.20998\)
Root analytic conductor: \(1.79164\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 402,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.511518410\)
\(L(\frac12)\) \(\approx\) \(2.511518410\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
67 \( 1 - T \)
good5 \( 1 - 0.710T + 5T^{2} \)
7 \( 1 + 1.81T + 7T^{2} \)
11 \( 1 - 0.578T + 11T^{2} \)
13 \( 1 - 4.72T + 13T^{2} \)
17 \( 1 - 4.20T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 5.44T + 23T^{2} \)
29 \( 1 + 4.72T + 29T^{2} \)
31 \( 1 + 2.18T + 31T^{2} \)
37 \( 1 + 8.07T + 37T^{2} \)
41 \( 1 - 9.96T + 41T^{2} \)
43 \( 1 + 3.75T + 43T^{2} \)
47 \( 1 + 2.05T + 47T^{2} \)
53 \( 1 - 4.71T + 53T^{2} \)
59 \( 1 - 0.132T + 59T^{2} \)
61 \( 1 + 6.15T + 61T^{2} \)
71 \( 1 - 8.15T + 71T^{2} \)
73 \( 1 + 5.49T + 73T^{2} \)
79 \( 1 - 2.35T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 3.83T + 89T^{2} \)
97 \( 1 - 6.78T + 97T^{2} \)
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

−11.35765940373097363302132884722, −10.32309940136743654687183489454, −9.533216631703754777808483679750, −8.478167869646061955457877620261, −7.48300789629183572127163979754, −6.31084883992519254884252387506, −5.63672200206299078418822852830, −4.03166901655056116949250672302, −3.31782319634588233175186642738, −1.82523912091582613325690118174, 1.82523912091582613325690118174, 3.31782319634588233175186642738, 4.03166901655056116949250672302, 5.63672200206299078418822852830, 6.31084883992519254884252387506, 7.48300789629183572127163979754, 8.478167869646061955457877620261, 9.533216631703754777808483679750, 10.32309940136743654687183489454, 11.35765940373097363302132884722

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