Properties

Label 2-966-23.6-c1-0-12
Degree $2$
Conductor $966$
Sign $0.847 - 0.531i$
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  + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.642 + 0.412i)5-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (−0.108 − 0.755i)10-s + (−0.850 − 1.86i)11-s + (−0.415 − 0.909i)12-s + (−0.0556 − 0.386i)13-s + (0.841 + 0.540i)14-s + (−0.732 + 0.215i)15-s + (−0.142 + 0.989i)16-s + (3.43 − 3.96i)17-s + ⋯
L(s)  = 1  + (−0.293 + 0.643i)2-s + (0.553 + 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.287 + 0.184i)5-s + (−0.267 + 0.308i)6-s + (0.0537 − 0.374i)7-s + (0.339 − 0.0996i)8-s + (0.280 + 0.180i)9-s + (−0.0343 − 0.238i)10-s + (−0.256 − 0.561i)11-s + (−0.119 − 0.262i)12-s + (−0.0154 − 0.107i)13-s + (0.224 + 0.144i)14-s + (−0.189 + 0.0555i)15-s + (−0.0355 + 0.247i)16-s + (0.833 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.847 - 0.531i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.847 - 0.531i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52365 + 0.438304i\)
\(L(\frac12)\) \(\approx\) \(1.52365 + 0.438304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
3 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (-4.64 + 1.18i)T \)
good5 \( 1 + (0.642 - 0.412i)T + (2.07 - 4.54i)T^{2} \)
11 \( 1 + (0.850 + 1.86i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.0556 + 0.386i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-3.43 + 3.96i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (-3.97 - 4.58i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-5.51 + 6.36i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (0.106 - 0.0312i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-3.54 - 2.27i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-0.356 + 0.229i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (2.90 + 0.854i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 8.25T + 47T^{2} \)
53 \( 1 + (1.20 - 8.37i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (1.10 + 7.69i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-2.23 + 0.657i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-2.03 + 4.45i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-1.10 + 2.42i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-3.41 - 3.93i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-0.214 - 1.49i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-9.64 - 6.20i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (13.7 + 4.03i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (1.44 - 0.931i)T + (40.2 - 88.2i)T^{2} \)
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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

−9.855702636730127076789601482594, −9.294954126535247057112202031061, −8.165822348979771159383247997943, −7.74220865942947196069557140787, −6.93800787056893158460552981552, −5.80613710284795965862556047370, −4.94152374310806167627101964489, −3.76952354023009328373178633482, −2.84087872021457808946414451815, −1.00627928982786694416843653849, 1.14859410582462735462652506618, 2.45889419997421644923701754699, 3.35643489941057899042973687842, 4.47302420182411259935336729055, 5.41476810734851799570880637676, 6.80309969555481111970775735338, 7.63544440473386883318697415067, 8.424466235709719231721904766211, 9.111372456485286813533002079158, 9.898630953296207293436418585300

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