Properties

Label 2-966-161.20-c1-0-9
Degree $2$
Conductor $966$
Sign $0.613 - 0.789i$
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  + (0.142 + 0.989i)2-s + (−0.909 − 0.415i)3-s + (−0.959 + 0.281i)4-s + (−0.108 + 0.125i)5-s + (0.281 − 0.959i)6-s + (−0.903 − 2.48i)7-s + (−0.415 − 0.909i)8-s + (0.654 + 0.755i)9-s + (−0.139 − 0.0897i)10-s + (−4.09 − 0.588i)11-s + (0.989 + 0.142i)12-s + (−3.02 + 4.71i)13-s + (2.33 − 1.24i)14-s + (0.150 − 0.0689i)15-s + (0.841 − 0.540i)16-s + (7.66 + 2.25i)17-s + ⋯
L(s)  = 1  + (0.100 + 0.699i)2-s + (−0.525 − 0.239i)3-s + (−0.479 + 0.140i)4-s + (−0.0486 + 0.0560i)5-s + (0.115 − 0.391i)6-s + (−0.341 − 0.939i)7-s + (−0.146 − 0.321i)8-s + (0.218 + 0.251i)9-s + (−0.0441 − 0.0283i)10-s + (−1.23 − 0.177i)11-s + (0.285 + 0.0410i)12-s + (−0.840 + 1.30i)13-s + (0.623 − 0.333i)14-s + (0.0389 − 0.0178i)15-s + (0.210 − 0.135i)16-s + (1.86 + 0.546i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.613 - 0.789i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.613 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02143 + 0.500070i\)
\(L(\frac12)\) \(\approx\) \(1.02143 + 0.500070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 - 0.989i)T \)
3 \( 1 + (0.909 + 0.415i)T \)
7 \( 1 + (0.903 + 2.48i)T \)
23 \( 1 + (4.73 + 0.791i)T \)
good5 \( 1 + (0.108 - 0.125i)T + (-0.711 - 4.94i)T^{2} \)
11 \( 1 + (4.09 + 0.588i)T + (10.5 + 3.09i)T^{2} \)
13 \( 1 + (3.02 - 4.71i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (-7.66 - 2.25i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-7.92 + 2.32i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-5.41 - 1.58i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-5.53 + 2.52i)T + (20.3 - 23.4i)T^{2} \)
37 \( 1 + (-8.33 + 7.22i)T + (5.26 - 36.6i)T^{2} \)
41 \( 1 + (-4.08 - 3.54i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (-2.31 - 1.05i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 - 0.473iT - 47T^{2} \)
53 \( 1 + (2.70 + 4.20i)T + (-22.0 + 48.2i)T^{2} \)
59 \( 1 + (3.18 - 4.95i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (2.08 + 4.57i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-8.24 + 1.18i)T + (64.2 - 18.8i)T^{2} \)
71 \( 1 + (-2.27 - 15.8i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-2.11 - 7.21i)T + (-61.4 + 39.4i)T^{2} \)
79 \( 1 + (-1.60 + 2.49i)T + (-32.8 - 71.8i)T^{2} \)
83 \( 1 + (-0.176 - 0.203i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-0.513 + 1.12i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (7.86 - 9.08i)T + (-13.8 - 96.0i)T^{2} \)
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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

−9.840596420896371240235409212526, −9.640023617696970506544831169314, −7.973926552888620694859562924177, −7.59673815841205973485107982238, −6.84876644818047414785486372278, −5.83268336728198922308771257656, −5.08022388591317401250461338067, −4.10569177193528526006478959094, −2.88279420678189294309529999088, −0.950034597595400463345296341459, 0.76944246923683498288344327130, 2.68383037634162488262130768348, 3.18655954025465735393651070496, 4.83059151767830546426235974783, 5.38772550995292620265017121936, 6.06694954914155071828745962532, 7.77950217981408026675175727854, 8.006164543125825356027089735812, 9.588797075003981185788171485137, 9.968068456873991570876693174530

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