Properties

Label 2-966-1.1-c1-0-1
Degree $2$
Conductor $966$
Sign $1$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3.70·5-s − 6-s − 7-s − 8-s + 9-s + 3.70·10-s + 12-s + 5.70·13-s + 14-s − 3.70·15-s + 16-s − 18-s − 5.40·19-s − 3.70·20-s − 21-s − 23-s − 24-s + 8.70·25-s − 5.70·26-s + 27-s − 28-s + 0.298·29-s + 3.70·30-s + 6·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.65·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.17·10-s + 0.288·12-s + 1.58·13-s + 0.267·14-s − 0.955·15-s + 0.250·16-s − 0.235·18-s − 1.23·19-s − 0.827·20-s − 0.218·21-s − 0.208·23-s − 0.204·24-s + 1.74·25-s − 1.11·26-s + 0.192·27-s − 0.188·28-s + 0.0554·29-s + 0.675·30-s + 1.07·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9772983172\)
\(L(\frac12)\) \(\approx\) \(0.9772983172\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 + T \)
good5 \( 1 + 3.70T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5.70T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.40T + 19T^{2} \)
29 \( 1 - 0.298T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 3.70T + 37T^{2} \)
41 \( 1 - 7.70T + 41T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 - 3.70T + 47T^{2} \)
53 \( 1 - 9.40T + 53T^{2} \)
59 \( 1 + 0.596T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 7.40T + 71T^{2} \)
73 \( 1 + 5.40T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 2.59T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + 1.10T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.983781874018989969865777907399, −8.820673214720440549220562579229, −8.452888141285712836172941424241, −7.75173931181191872057782594951, −6.89253921530535890639520588773, −6.01478067007850531862752235801, −4.26420864617901539745847991860, −3.73918192404056483558940036534, −2.59454545221232768006274794575, −0.843104700239064260327536674643, 0.843104700239064260327536674643, 2.59454545221232768006274794575, 3.73918192404056483558940036534, 4.26420864617901539745847991860, 6.01478067007850531862752235801, 6.89253921530535890639520588773, 7.75173931181191872057782594951, 8.452888141285712836172941424241, 8.820673214720440549220562579229, 9.983781874018989969865777907399

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