Properties

Label 2-966-161.18-c1-0-4
Degree $2$
Conductor $966$
Sign $-0.718 - 0.695i$
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  + (0.235 + 0.971i)2-s + (−0.327 − 0.945i)3-s + (−0.888 + 0.458i)4-s + (−0.262 + 0.105i)5-s + (0.841 − 0.540i)6-s + (−1.94 − 1.79i)7-s + (−0.654 − 0.755i)8-s + (−0.786 + 0.618i)9-s + (−0.164 − 0.230i)10-s + (0.182 − 0.751i)11-s + (0.723 + 0.690i)12-s + (−0.0379 + 0.0830i)13-s + (1.28 − 2.31i)14-s + (0.185 + 0.214i)15-s + (0.580 − 0.814i)16-s + (0.140 + 2.94i)17-s + ⋯
L(s)  = 1  + (0.166 + 0.687i)2-s + (−0.188 − 0.545i)3-s + (−0.444 + 0.229i)4-s + (−0.117 + 0.0470i)5-s + (0.343 − 0.220i)6-s + (−0.735 − 0.677i)7-s + (−0.231 − 0.267i)8-s + (−0.262 + 0.206i)9-s + (−0.0519 − 0.0729i)10-s + (0.0549 − 0.226i)11-s + (0.208 + 0.199i)12-s + (−0.0105 + 0.0230i)13-s + (0.343 − 0.618i)14-s + (0.0478 + 0.0552i)15-s + (0.145 − 0.203i)16-s + (0.0340 + 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.718 - 0.695i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (823, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.718 - 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270502 + 0.668507i\)
\(L(\frac12)\) \(\approx\) \(0.270502 + 0.668507i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.235 - 0.971i)T \)
3 \( 1 + (0.327 + 0.945i)T \)
7 \( 1 + (1.94 + 1.79i)T \)
23 \( 1 + (-1.20 - 4.64i)T \)
good5 \( 1 + (0.262 - 0.105i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (-0.182 + 0.751i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (0.0379 - 0.0830i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-0.140 - 2.94i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (0.369 - 7.76i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (1.47 - 0.947i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-2.38 + 0.460i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (1.53 - 1.20i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (0.0592 - 0.412i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (2.78 - 3.21i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (3.32 + 5.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.993 + 0.0949i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (1.73 + 2.43i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (3.39 - 9.81i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (-8.56 + 8.16i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (-0.539 - 0.158i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (12.3 - 6.35i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (-12.1 + 1.15i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (-1.85 - 12.9i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (17.6 + 3.40i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (1.50 - 10.4i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.18649161151580195999452738362, −9.523633362102189446495101151334, −8.344287494260656792835847992557, −7.72655956459415696809200726509, −6.91954634610507146011027729807, −6.12632069087971552707154759233, −5.44065989001585857072071300205, −4.03015800387294298716786912361, −3.32750118031336480530073447168, −1.48966459386313645713621494414, 0.32833166492550307483320834942, 2.37737028355319045588659351830, 3.16464062859943334190671322042, 4.38095263274526185052131717883, 5.08330242982784620507698304921, 6.14382977128158947436503310899, 7.00481331829435620622615433435, 8.371215368118716779304558308536, 9.168448488847325074094450614096, 9.688846172805594407234538151747

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