Properties

Label 2-966-23.8-c1-0-2
Degree $2$
Conductor $966$
Sign $0.703 - 0.710i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−1.94 − 2.24i)5-s + (−0.959 + 0.281i)6-s + (−0.841 − 0.540i)7-s + (−0.415 + 0.909i)8-s + (−0.654 + 0.755i)9-s + (−2.49 + 1.60i)10-s + (0.361 + 2.51i)11-s + (0.142 + 0.989i)12-s + (−5.77 + 3.71i)13-s + (−0.654 + 0.755i)14-s + (−1.23 + 2.70i)15-s + (0.841 + 0.540i)16-s + (3.79 − 1.11i)17-s + ⋯
L(s)  = 1  + (0.100 − 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.869 − 1.00i)5-s + (−0.391 + 0.115i)6-s + (−0.317 − 0.204i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.790 + 0.507i)10-s + (0.109 + 0.758i)11-s + (0.0410 + 0.285i)12-s + (−1.60 + 1.02i)13-s + (−0.175 + 0.201i)14-s + (−0.318 + 0.697i)15-s + (0.210 + 0.135i)16-s + (0.920 − 0.270i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.703 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.260491 + 0.108606i\)
\(L(\frac12)\) \(\approx\) \(0.260491 + 0.108606i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-3.74 - 2.99i)T \)
good5 \( 1 + (1.94 + 2.24i)T + (-0.711 + 4.94i)T^{2} \)
11 \( 1 + (-0.361 - 2.51i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (5.77 - 3.71i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-3.79 + 1.11i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (0.341 + 0.100i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (5.47 - 1.60i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-3.86 + 8.45i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (2.60 - 3.00i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (-1.43 - 1.65i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (1.87 + 4.11i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + 7.94T + 47T^{2} \)
53 \( 1 + (-10.3 - 6.66i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (5.59 - 3.59i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-0.303 + 0.665i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (0.906 - 6.30i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (1.57 - 10.9i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-2.57 - 0.756i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (13.8 - 8.92i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (6.34 - 7.32i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-3.81 - 8.35i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (6.69 + 7.72i)T + (-13.8 + 96.0i)T^{2} \)
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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

−9.912612562123857824049756566847, −9.523672766618908769804922409580, −8.517552165864939772226669595660, −7.51248105011570797358363379489, −7.03045900552377780120716329574, −5.51745873693415581678534408069, −4.70108693202045027328838744076, −3.97585550604194067205726735046, −2.56540555133614828188245399553, −1.27827320724743924822280510612, 0.14157340313262724666983133414, 2.97582166271242539182487615680, 3.47848342851728486583546211521, 4.78416924887711669526573365611, 5.57631388347087640563715570982, 6.56475535303510286334633301193, 7.36804303323051438798581213610, 8.040994066070368632476822970520, 9.003989291139736095326284666467, 10.06156546613963202367474614524

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