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Properties

Label 2-966-161.10-c1-0-25
Degree $2$
Conductor $966$
Sign $-0.115 + 0.993i$
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 + (4.16 − 0.397i)5-s + (−0.989 + 0.142i)6-s + (0.0805 + 2.64i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (0.986 − 4.06i)10-s + (−0.191 + 0.0663i)11-s + (−0.189 + 0.981i)12-s + (−1.51 − 5.17i)13-s + (2.52 + 0.788i)14-s + (−2.26 − 3.52i)15-s + (0.235 + 0.971i)16-s + (−4.53 − 1.81i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (−0.264 − 0.513i)3-s + (−0.393 − 0.309i)4-s + (1.86 − 0.177i)5-s + (−0.404 + 0.0580i)6-s + (0.0304 + 0.999i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (0.312 − 1.28i)10-s + (−0.0577 + 0.0199i)11-s + (−0.0546 + 0.283i)12-s + (−0.421 − 1.43i)13-s + (0.674 + 0.210i)14-s + (−0.584 − 0.909i)15-s + (0.0589 + 0.242i)16-s + (−1.09 − 0.439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.115 + 0.993i$
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: \((2,\ 966,\ (\ :1/2),\ -0.115 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40353 - 1.57614i\)
\(L(\frac12)\) \(\approx\) \(1.40353 - 1.57614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0805 - 2.64i)T \)
23 \( 1 + (-4.79 - 0.0590i)T \)
good5 \( 1 + (-4.16 + 0.397i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (0.191 - 0.0663i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.51 + 5.17i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (4.53 + 1.81i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-3.26 + 1.30i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.707 + 4.91i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-7.82 + 0.372i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-0.879 - 0.626i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (2.62 + 1.19i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-3.34 + 5.20i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-6.61 + 3.81i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.18 - 5.43i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (9.75 + 2.36i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (5.15 + 2.65i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-2.09 - 10.8i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (0.0174 + 0.0201i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (6.57 - 8.35i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (11.2 - 11.7i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-2.10 - 4.61i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.851 - 17.8i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-0.146 + 0.319i)T + (-63.5 - 73.3i)T^{2} \)
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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

−9.850047583490697204394387924670, −9.157310918321573950048726752941, −8.427536209016755234714463516721, −7.05485132944321026551131966206, −6.02610783133198924828183897899, −5.48374741153471837107504663861, −4.83282920147797419995459804031, −2.70307141083735519411170467899, −2.44998082761253187135990805081, −1.04152375191357281607644659567, 1.54521121956127400412314571178, 2.97058208040942605884474541146, 4.41790060266786597967338410328, 4.96938010183692141537760458437, 6.11737633422254186102237775296, 6.60302939096078644575704072954, 7.41014134055542205028763651078, 8.943079773817086416515425759613, 9.296989870655800868987581621995, 10.20002275987643518175222047544

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