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Properties

Label 2-966-23.16-c1-0-6
Degree $2$
Conductor $966$
Sign $0.997 - 0.0647i$
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  + (−0.654 + 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (−0.129 − 0.283i)5-s + (0.142 − 0.989i)6-s + (−0.959 + 0.281i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (0.298 + 0.0877i)10-s + (0.0711 + 0.0821i)11-s + (0.654 + 0.755i)12-s + (−2.11 − 0.622i)13-s + (0.415 − 0.909i)14-s + (0.262 + 0.168i)15-s + (−0.959 + 0.281i)16-s + (−0.200 + 1.39i)17-s + ⋯
L(s)  = 1  + (−0.463 + 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.0578 − 0.126i)5-s + (0.0580 − 0.404i)6-s + (−0.362 + 0.106i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (0.0945 + 0.0277i)10-s + (0.0214 + 0.0247i)11-s + (0.189 + 0.218i)12-s + (−0.587 − 0.172i)13-s + (0.111 − 0.243i)14-s + (0.0676 + 0.0434i)15-s + (−0.239 + 0.0704i)16-s + (−0.0486 + 0.338i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.997 - 0.0647i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.997 - 0.0647i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.857459 + 0.0277906i\)
\(L(\frac12)\) \(\approx\) \(0.857459 + 0.0277906i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 0.755i)T \)
3 \( 1 + (0.841 - 0.540i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (-4.68 + 1.04i)T \)
good5 \( 1 + (0.129 + 0.283i)T + (-3.27 + 3.77i)T^{2} \)
11 \( 1 + (-0.0711 - 0.0821i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (2.11 + 0.622i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.200 - 1.39i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (0.126 + 0.876i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.257 + 1.78i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (1.56 + 1.00i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-0.477 + 1.04i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (2.45 + 5.38i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (1.41 - 0.909i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + (-12.4 + 3.65i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-11.5 - 3.39i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (1.74 + 1.12i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (-3.52 + 4.06i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (-1.85 + 2.13i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-0.444 - 3.09i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (2.50 + 0.736i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (3.87 - 8.47i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-7.55 + 4.85i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (6.31 + 13.8i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.06546217027676150546814236784, −9.112138451398210349356071050057, −8.514970534701462080508031791065, −7.35707211799999653977014512778, −6.71575211367160317071331992259, −5.73959404469755786731623938017, −4.97641694827581343535649460862, −3.93233219358181611495937694130, −2.46437919348870338481391513132, −0.64865147950710043763825083231, 0.991005363331114507171312770226, 2.41901871248383906825735520858, 3.49308693487481138871509631107, 4.71985652794296252143179046894, 5.67533646891094502919338487144, 6.96076268061501095260322023456, 7.27548127249145314173410953506, 8.495282220349402339012255309376, 9.271615806625412418418722897189, 10.09828198203810288117576572445

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