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Properties

Label 2-968-968.131-c1-0-111
Degree $2$
Conductor $968$
Sign $0.242 + 0.970i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
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  + (0.764 − 1.18i)2-s + 3.40·3-s + (−0.830 − 1.81i)4-s + (2.60 − 4.04i)6-s + (−2.79 − 0.402i)8-s + 8.58·9-s + (1.82 + 2.76i)11-s + (−2.82 − 6.19i)12-s + (−2.61 + 3.02i)16-s + (−1.89 − 6.47i)17-s + (6.56 − 10.2i)18-s + (−0.897 + 3.05i)19-s + (4.69 − 0.0568i)22-s + (−9.52 − 1.37i)24-s + (−4.79 + 1.40i)25-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)2-s + 1.96·3-s + (−0.415 − 0.909i)4-s + (1.06 − 1.65i)6-s + (−0.989 − 0.142i)8-s + 2.86·9-s + (0.550 + 0.834i)11-s + (−0.816 − 1.78i)12-s + (−0.654 + 0.755i)16-s + (−0.460 − 1.56i)17-s + (1.54 − 2.40i)18-s + (−0.206 + 0.701i)19-s + (0.999 − 0.0121i)22-s + (−1.94 − 0.279i)24-s + (−0.959 + 0.281i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.06914 - 2.39648i\)
\(L(\frac12)\) \(\approx\) \(3.06914 - 2.39648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.764 + 1.18i)T \)
11 \( 1 + (-1.82 - 2.76i)T \)
good3 \( 1 - 3.40T + 3T^{2} \)
5 \( 1 + (4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.996 + 6.92i)T^{2} \)
13 \( 1 + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (1.89 + 6.47i)T + (-14.3 + 9.19i)T^{2} \)
19 \( 1 + (0.897 - 3.05i)T + (-15.9 - 10.2i)T^{2} \)
23 \( 1 + (3.27 + 22.7i)T^{2} \)
29 \( 1 + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (20.3 + 23.4i)T^{2} \)
37 \( 1 + (24.2 + 27.9i)T^{2} \)
41 \( 1 + (1.55 - 2.41i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (10.8 + 1.55i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (-19.5 + 42.7i)T^{2} \)
53 \( 1 + (7.54 - 52.4i)T^{2} \)
59 \( 1 + (12.9 - 8.30i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (-11.7 - 7.56i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (-59.7 - 38.3i)T^{2} \)
73 \( 1 + (3.95 + 3.42i)T + (10.3 + 72.2i)T^{2} \)
79 \( 1 + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (2.13 - 1.85i)T + (11.8 - 82.1i)T^{2} \)
89 \( 1 + (-15.4 + 4.52i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (2.50 - 17.4i)T + (-93.0 - 27.3i)T^{2} \)
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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

−9.703960895687155472006237053226, −9.247758435196700538105215901883, −8.416741844652304245330883437278, −7.42500177785831850684553211892, −6.61287253081951283196354584622, −4.98647652172477261783563152329, −4.14665718633151212434743241726, −3.37102671281642031757359974867, −2.40452153984876150363176253094, −1.58706020827627933060430988291, 1.93251875355976135151729338761, 3.16854840683958945523853652751, 3.81896581436069037526059388414, 4.63378972185478509585263349829, 6.15709757452906754065270349967, 6.87724512643702623884185902229, 7.926471760005415648760708855728, 8.362056075123103786039696665193, 9.011486729414420888078187298994, 9.741114110062866841711729150805

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