Properties

Label 2-35-1.1-c1-0-0
Degree $2$
Conductor $35$
Sign $1$
Analytic cond. $0.279476$
Root an. cond. $0.528655$
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  − 2.56·2-s + 1.56·3-s + 4.56·4-s + 5-s − 4·6-s − 7-s − 6.56·8-s − 0.561·9-s − 2.56·10-s − 1.56·11-s + 7.12·12-s + 0.438·13-s + 2.56·14-s + 1.56·15-s + 7.68·16-s − 0.438·17-s + 1.43·18-s − 7.12·19-s + 4.56·20-s − 1.56·21-s + 4·22-s + 3.12·23-s − 10.2·24-s + 25-s − 1.12·26-s − 5.56·27-s − 4.56·28-s + ⋯
L(s)  = 1  − 1.81·2-s + 0.901·3-s + 2.28·4-s + 0.447·5-s − 1.63·6-s − 0.377·7-s − 2.31·8-s − 0.187·9-s − 0.810·10-s − 0.470·11-s + 2.05·12-s + 0.121·13-s + 0.684·14-s + 0.403·15-s + 1.92·16-s − 0.106·17-s + 0.339·18-s − 1.63·19-s + 1.01·20-s − 0.340·21-s + 0.852·22-s + 0.651·23-s − 2.09·24-s + 0.200·25-s − 0.220·26-s − 1.07·27-s − 0.862·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.279476\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4600763520\)
\(L(\frac12)\) \(\approx\) \(0.4600763520\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 + T \)
good2 \( 1 + 2.56T + 2T^{2} \)
3 \( 1 - 1.56T + 3T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
13 \( 1 - 0.438T + 13T^{2} \)
17 \( 1 + 0.438T + 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 5.12T + 41T^{2} \)
43 \( 1 - 0.876T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 + 5.12T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 2.43T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 1.12T + 89T^{2} \)
97 \( 1 - 5.80T + 97T^{2} \)
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

−16.90322846331634953577134877295, −15.74352143908137457099974538480, −14.58335792317703476215761000619, −12.96793398602071777540469125017, −11.12711409504629322506872055031, −9.982496960603323626825608149426, −8.914097901089898875492127200173, −8.054746333088715318280391070413, −6.48150256884439156034400207857, −2.53657717653485394345612541605, 2.53657717653485394345612541605, 6.48150256884439156034400207857, 8.054746333088715318280391070413, 8.914097901089898875492127200173, 9.982496960603323626825608149426, 11.12711409504629322506872055031, 12.96793398602071777540469125017, 14.58335792317703476215761000619, 15.74352143908137457099974538480, 16.90322846331634953577134877295

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