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Properties

Label 2-966-161.10-c1-0-28
Degree $2$
Conductor $966$
Sign $-0.977 - 0.210i$
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  + (0.327 − 0.945i)2-s + (0.458 + 0.888i)3-s + (−0.786 − 0.618i)4-s + (−1.27 + 0.121i)5-s + (0.989 − 0.142i)6-s + (−0.678 + 2.55i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.302 + 1.24i)10-s + (−1.09 + 0.380i)11-s + (0.189 − 0.981i)12-s + (−1.87 − 6.37i)13-s + (2.19 + 1.47i)14-s + (−0.693 − 1.07i)15-s + (0.235 + 0.971i)16-s + (0.0653 + 0.0261i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (−0.571 + 0.0545i)5-s + (0.404 − 0.0580i)6-s + (−0.256 + 0.966i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.0956 + 0.394i)10-s + (−0.331 + 0.114i)11-s + (0.0546 − 0.283i)12-s + (−0.519 − 1.76i)13-s + (0.586 + 0.394i)14-s + (−0.179 − 0.278i)15-s + (0.0589 + 0.242i)16-s + (0.0158 + 0.00634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.977 - 0.210i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.977 - 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00715958 + 0.0673010i\)
\(L(\frac12)\) \(\approx\) \(0.00715958 + 0.0673010i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.327 + 0.945i)T \)
3 \( 1 + (-0.458 - 0.888i)T \)
7 \( 1 + (0.678 - 2.55i)T \)
23 \( 1 + (0.864 + 4.71i)T \)
good5 \( 1 + (1.27 - 0.121i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (1.09 - 0.380i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.87 + 6.37i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.0653 - 0.0261i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (3.05 - 1.22i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.347 - 2.41i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (2.40 - 0.114i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (9.68 + 6.89i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (6.49 + 2.96i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (4.18 - 6.51i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-1.08 + 0.628i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.70 - 1.78i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-14.5 - 3.53i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-7.98 - 4.11i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.156 - 0.813i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-3.06 - 3.53i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (1.16 - 1.47i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (11.8 - 12.4i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (2.25 + 4.92i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.0883 - 1.85i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-3.67 + 8.03i)T + (-63.5 - 73.3i)T^{2} \)
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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

−9.882144053124298745685471412438, −8.626229960824212770538129688929, −8.313729964101304964286884774106, −7.11465387110313013014048634651, −5.69978140787089497393285078815, −5.18489095873057399587672892433, −3.98084659331970810923433193374, −3.10094259394875831414749043509, −2.23592426715665195602102039452, −0.02624802339623618242754247880, 1.89858018621979962780088855811, 3.51668146596051145793951016624, 4.21341454636615972198223855727, 5.24258295293047397360889899170, 6.66394421563571644093063779422, 6.92045172593026062797342416800, 7.82529500311166714075418127691, 8.551445835089655471938318340410, 9.490799920443516579335746461377, 10.31967168154167157812281217104

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