Properties

Label 2-966-1.1-c3-0-37
Degree $2$
Conductor $966$
Sign $1$
Analytic cond. $56.9958$
Root an. cond. $7.54955$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s + 13.7·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s + 27.4·10-s + 45.9·11-s − 12·12-s + 5.45·13-s + 14·14-s − 41.1·15-s + 16·16-s + 39.2·17-s + 18·18-s + 27.3·19-s + 54.9·20-s − 21·21-s + 91.8·22-s − 23·23-s − 24·24-s + 63.5·25-s + 10.9·26-s − 27·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.22·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.868·10-s + 1.25·11-s − 0.288·12-s + 0.116·13-s + 0.267·14-s − 0.709·15-s + 0.250·16-s + 0.560·17-s + 0.235·18-s + 0.330·19-s + 0.614·20-s − 0.218·21-s + 0.889·22-s − 0.208·23-s − 0.204·24-s + 0.508·25-s + 0.0822·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(56.9958\)
Root analytic conductor: \(7.54955\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.247061004\)
\(L(\frac12)\) \(\approx\) \(4.247061004\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + 3T \)
7 \( 1 - 7T \)
23 \( 1 + 23T \)
good5 \( 1 - 13.7T + 125T^{2} \)
11 \( 1 - 45.9T + 1.33e3T^{2} \)
13 \( 1 - 5.45T + 2.19e3T^{2} \)
17 \( 1 - 39.2T + 4.91e3T^{2} \)
19 \( 1 - 27.3T + 6.85e3T^{2} \)
29 \( 1 - 9.12T + 2.43e4T^{2} \)
31 \( 1 - 29.6T + 2.97e4T^{2} \)
37 \( 1 - 350.T + 5.06e4T^{2} \)
41 \( 1 + 445.T + 6.89e4T^{2} \)
43 \( 1 + 447.T + 7.95e4T^{2} \)
47 \( 1 - 585.T + 1.03e5T^{2} \)
53 \( 1 + 595.T + 1.48e5T^{2} \)
59 \( 1 + 365.T + 2.05e5T^{2} \)
61 \( 1 - 629.T + 2.26e5T^{2} \)
67 \( 1 + 150.T + 3.00e5T^{2} \)
71 \( 1 + 125.T + 3.57e5T^{2} \)
73 \( 1 + 702.T + 3.89e5T^{2} \)
79 \( 1 - 783.T + 4.93e5T^{2} \)
83 \( 1 - 744.T + 5.71e5T^{2} \)
89 \( 1 + 40.6T + 7.04e5T^{2} \)
97 \( 1 - 394.T + 9.12e5T^{2} \)
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

−9.808301389746525304474184551762, −8.979276058752895934320223806555, −7.77992650524409667239707981206, −6.67941682573029616787829204164, −6.12606743677945289657751299493, −5.36394958771885761063335484562, −4.49724108013594244841814373051, −3.38001188915950982271562424579, −1.99546644446973938367685688624, −1.14099143173492481957862385567, 1.14099143173492481957862385567, 1.99546644446973938367685688624, 3.38001188915950982271562424579, 4.49724108013594244841814373051, 5.36394958771885761063335484562, 6.12606743677945289657751299493, 6.67941682573029616787829204164, 7.77992650524409667239707981206, 8.979276058752895934320223806555, 9.808301389746525304474184551762

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