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Properties

Label 2-966-161.10-c1-0-24
Degree $2$
Conductor $966$
Sign $0.825 - 0.564i$
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 + (3.17 − 0.303i)5-s + (−0.989 + 0.142i)6-s + (1.76 − 1.96i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.752 + 3.10i)10-s + (0.0727 − 0.0251i)11-s + (0.189 − 0.981i)12-s + (−1.09 − 3.73i)13-s + (1.28 + 2.31i)14-s + (1.72 + 2.68i)15-s + (0.235 + 0.971i)16-s + (0.538 + 0.215i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (1.42 − 0.135i)5-s + (−0.404 + 0.0580i)6-s + (0.667 − 0.744i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.237 + 0.980i)10-s + (0.0219 − 0.00759i)11-s + (0.0546 − 0.283i)12-s + (−0.303 − 1.03i)13-s + (0.343 + 0.618i)14-s + (0.445 + 0.693i)15-s + (0.0589 + 0.242i)16-s + (0.130 + 0.0522i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.825 - 0.564i$
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: \((2,\ 966,\ (\ :1/2),\ 0.825 - 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95400 + 0.604763i\)
\(L(\frac12)\) \(\approx\) \(1.95400 + 0.604763i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.76 + 1.96i)T \)
23 \( 1 + (-4.76 - 0.505i)T \)
good5 \( 1 + (-3.17 + 0.303i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-0.0727 + 0.0251i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.09 + 3.73i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.538 - 0.215i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-1.86 + 0.745i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.420 + 2.92i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (2.53 - 0.120i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (2.39 + 1.70i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-0.862 - 0.393i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-1.60 + 2.49i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (3.12 - 1.80i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.840 - 0.881i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-5.61 - 1.36i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-4.00 - 2.06i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-2.43 - 12.6i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-0.0597 - 0.0689i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (9.57 - 12.1i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (3.57 - 3.74i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-3.62 - 7.94i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.340 - 7.15i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (7.48 - 16.3i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.02008041718618491808840499211, −9.305034216466529934040216018217, −8.482067395467949962503030504642, −7.60689796490499690028286767728, −6.76861106924371955642864876042, −5.50144349926378531487410886394, −5.24244513841906396815049608467, −4.01200778170021106726186942907, −2.60109788823235114168979231109, −1.17632417057073737846809446998, 1.51855411005735543303321055191, 2.14254177474380798130188399936, 3.15491022579049280605605462439, 4.74556076336769714659223233000, 5.56300909560966455355147126305, 6.53757377825044481513183584416, 7.46930565485141035417101062457, 8.600474765461153671597458882985, 9.184507056615273241123153669236, 9.776349352859589038862549164231

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