Properties

Label 1-67-67.66-r1-0-0
Degree $1$
Conductor $67$
Sign $1$
Analytic cond. $7.20014$
Root an. cond. $7.20014$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 13-s + 14-s + 15-s + 16-s + 17-s − 18-s + 19-s − 20-s + 21-s + 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $1$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{67} (66, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3729296206\)
\(L(\frac12)\) \(\approx\) \(0.3729296206\)
\(L(1)\) \(\approx\) \(0.3838066288\)
\(L(1)\) \(\approx\) \(0.3838066288\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−31.80869018654966570048879528906, −30.2983684160728915396578120066, −29.03735518638643006914975182318, −28.62224652761012483734243629966, −27.2611308641117556922049508591, −26.70174569224816843100029768924, −25.25823792241517996178420996375, −23.91332809292266679012770993527, −23.089595227482036622305087852069, −21.748513425850136642643612749868, −20.24915647424801226494257358160, −19.089227625032628781940265850209, −18.3437356923891410539292266510, −16.86174789733574687551804808301, −16.14199851379843707740194080055, −15.220716962584560554285096203369, −12.68248175206236824036542572352, −11.85025250054448170584466652275, −10.59843546331941620439755468854, −9.613018248078095366246116971147, −7.760383041884180603322129774131, −6.8856313660388414445999220040, −5.27458723627780946882402310139, −3.12142075096677987543258628564, −0.60431475891438510756499008752, 0.60431475891438510756499008752, 3.12142075096677987543258628564, 5.27458723627780946882402310139, 6.8856313660388414445999220040, 7.760383041884180603322129774131, 9.613018248078095366246116971147, 10.59843546331941620439755468854, 11.85025250054448170584466652275, 12.68248175206236824036542572352, 15.220716962584560554285096203369, 16.14199851379843707740194080055, 16.86174789733574687551804808301, 18.3437356923891410539292266510, 19.089227625032628781940265850209, 20.24915647424801226494257358160, 21.748513425850136642643612749868, 23.089595227482036622305087852069, 23.91332809292266679012770993527, 25.25823792241517996178420996375, 26.70174569224816843100029768924, 27.2611308641117556922049508591, 28.62224652761012483734243629966, 29.03735518638643006914975182318, 30.2983684160728915396578120066, 31.80869018654966570048879528906

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