Properties

Label 2-966-1.1-c3-0-0
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s − 12.7·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 25.5·10-s − 4.56·11-s − 12·12-s − 70.6·13-s + 14·14-s + 38.3·15-s + 16·16-s − 78.3·17-s − 18·18-s − 43.1·19-s − 51.1·20-s + 21·21-s + 9.12·22-s − 23·23-s + 24·24-s + 38.3·25-s + 141.·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.14·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.808·10-s − 0.125·11-s − 0.288·12-s − 1.50·13-s + 0.267·14-s + 0.660·15-s + 0.250·16-s − 1.11·17-s − 0.235·18-s − 0.521·19-s − 0.571·20-s + 0.218·21-s + 0.0884·22-s − 0.208·23-s + 0.204·24-s + 0.307·25-s + 1.06·26-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

\(L(2)\) \(\approx\) \(0.03501366118\)
\(L(\frac12)\) \(\approx\) \(0.03501366118\)
\(L(\frac{5}{2})\) 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 \( 1 + 12.7T + 125T^{2} \)
11 \( 1 + 4.56T + 1.33e3T^{2} \)
13 \( 1 + 70.6T + 2.19e3T^{2} \)
17 \( 1 + 78.3T + 4.91e3T^{2} \)
19 \( 1 + 43.1T + 6.85e3T^{2} \)
29 \( 1 - 166.T + 2.43e4T^{2} \)
31 \( 1 + 147.T + 2.97e4T^{2} \)
37 \( 1 + 363.T + 5.06e4T^{2} \)
41 \( 1 + 295.T + 6.89e4T^{2} \)
43 \( 1 + 456.T + 7.95e4T^{2} \)
47 \( 1 + 117.T + 1.03e5T^{2} \)
53 \( 1 + 221.T + 1.48e5T^{2} \)
59 \( 1 + 22.7T + 2.05e5T^{2} \)
61 \( 1 + 399.T + 2.26e5T^{2} \)
67 \( 1 - 446.T + 3.00e5T^{2} \)
71 \( 1 - 321.T + 3.57e5T^{2} \)
73 \( 1 + 519.T + 3.89e5T^{2} \)
79 \( 1 - 85.1T + 4.93e5T^{2} \)
83 \( 1 + 163.T + 5.71e5T^{2} \)
89 \( 1 + 955.T + 7.04e5T^{2} \)
97 \( 1 + 476.T + 9.12e5T^{2} \)
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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

−9.744395233796035301212770589491, −8.720894512365698480086632147369, −7.993863698147338260624729006286, −7.03909071940489186211916395090, −6.63164466820457276007241104807, −5.22805145584909756714635375429, −4.37435964369630439039932267253, −3.20500551258018533160326695377, −1.91780160769727424060674135603, −0.10857404180743991762080103180, 0.10857404180743991762080103180, 1.91780160769727424060674135603, 3.20500551258018533160326695377, 4.37435964369630439039932267253, 5.22805145584909756714635375429, 6.63164466820457276007241104807, 7.03909071940489186211916395090, 7.993863698147338260624729006286, 8.720894512365698480086632147369, 9.744395233796035301212770589491

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