Properties

Label 2-966-23.8-c1-0-19
Degree $2$
Conductor $966$
Sign $0.379 + 0.925i$
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 + (−0.150 − 0.173i)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 + (−0.193 + 0.124i)10-s + (−0.117 − 0.820i)11-s + (−0.142 − 0.989i)12-s + (2.13 − 1.37i)13-s + (−0.654 + 0.755i)14-s + (0.0955 − 0.209i)15-s + (0.841 + 0.540i)16-s + (3.16 − 0.928i)17-s + ⋯
L(s)  = 1  + (0.100 − 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (−0.0673 − 0.0777i)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 + (−0.0611 + 0.0393i)10-s + (−0.0355 − 0.247i)11-s + (−0.0410 − 0.285i)12-s + (0.593 − 0.381i)13-s + (−0.175 + 0.201i)14-s + (0.0246 − 0.0540i)15-s + (0.210 + 0.135i)16-s + (0.767 − 0.225i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.379 + 0.925i$
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: \((2,\ 966,\ (\ :1/2),\ 0.379 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36144 - 0.913457i\)
\(L(\frac12)\) \(\approx\) \(1.36144 - 0.913457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.22 + 2.27i)T \)
good5 \( 1 + (0.150 + 0.173i)T + (-0.711 + 4.94i)T^{2} \)
11 \( 1 + (0.117 + 0.820i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (-2.13 + 1.37i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-3.16 + 0.928i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (-1.67 - 0.492i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (2.74 - 0.805i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-2.53 + 5.54i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-2.36 + 2.72i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (1.88 + 2.17i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-2.86 - 6.27i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 - 8.09T + 47T^{2} \)
53 \( 1 + (-1.19 - 0.767i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (3.59 - 2.30i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-3.88 + 8.49i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (-1.30 + 9.09i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (0.768 - 5.34i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-0.777 - 0.228i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (0.112 - 0.0720i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (2.71 - 3.13i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (2.22 + 4.86i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (2.78 + 3.21i)T + (-13.8 + 96.0i)T^{2} \)
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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

−9.912436323078967870742392910722, −9.235975056973070779272539351728, −8.379745488426142707726354142127, −7.55270671967024583695331876038, −6.24055818654268864957520377288, −5.35036194198942204450151100375, −4.31551160338519487038116435708, −3.45264486435376465406275569442, −2.56508142863981499449929702674, −0.849746953267475281136798311338, 1.30252229582268094167934264366, 2.93482893101183574189185918114, 3.87523347931753573633782385834, 5.18908825308213218934875441398, 5.94870419705976171548111852687, 6.95390550384376357546778594343, 7.44545677192873537714616234730, 8.474776517138949185912411395924, 9.116548495753436147316285960204, 9.949382258524240775173396499041

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