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Properties

Label 2-966-161.10-c1-0-29
Degree $2$
Conductor $966$
Sign $-0.970 + 0.243i$
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 + (0.332 − 0.0317i)5-s + (0.989 − 0.142i)6-s + (−1.62 − 2.08i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (0.0787 − 0.324i)10-s + (−2.53 + 0.875i)11-s + (0.189 − 0.981i)12-s + (−1.26 − 4.30i)13-s + (−2.50 + 0.857i)14-s + (0.180 + 0.280i)15-s + (0.235 + 0.971i)16-s + (−3.66 − 1.46i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (0.148 − 0.0141i)5-s + (0.404 − 0.0580i)6-s + (−0.615 − 0.788i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (0.0248 − 0.102i)10-s + (−0.762 + 0.264i)11-s + (0.0546 − 0.283i)12-s + (−0.350 − 1.19i)13-s + (−0.668 + 0.229i)14-s + (0.0466 + 0.0725i)15-s + (0.0589 + 0.242i)16-s + (−0.888 − 0.355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.243i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.970 + 0.243i$
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: \((2,\ 966,\ (\ :1/2),\ -0.970 + 0.243i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0985146 - 0.798342i\)
\(L(\frac12)\) \(\approx\) \(0.0985146 - 0.798342i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 + 2.08i)T \)
23 \( 1 + (1.34 - 4.60i)T \)
good5 \( 1 + (-0.332 + 0.0317i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (2.53 - 0.875i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.26 + 4.30i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (3.66 + 1.46i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-4.04 + 1.62i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (1.19 + 8.29i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (6.30 - 0.300i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (2.82 + 2.01i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-3.26 - 1.49i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.08 + 3.23i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-8.79 + 5.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.557 - 0.585i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (5.90 + 1.43i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (5.30 + 2.73i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (2.06 + 10.7i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (0.467 + 0.539i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (2.16 - 2.75i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-5.07 + 5.32i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-4.18 - 9.15i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.172 - 3.62i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (2.51 - 5.49i)T + (-63.5 - 73.3i)T^{2} \)
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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

−9.660350287423783530518030598118, −9.283666995607449660022461227352, −7.87035037605501963627581311053, −7.33769159272678268242337591395, −5.87776885932244271763733398803, −5.13922025254160140129076785740, −4.06298943108862232070187781097, −3.24163655248252543304618042826, −2.22844130133467515236153246653, −0.30869382121830346348622636290, 2.00990268010142812277540221583, 3.06783870206853560486658026179, 4.28380030503374233180526934556, 5.46654217017754271501430143466, 6.15481578526265782911430750310, 7.00113995075521570900342628963, 7.74080559653996941625717920574, 8.850072029009741784614088271848, 9.151259821159154893000102638128, 10.25042834905340103265747802552

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