Properties

Label 2-966-23.4-c1-0-14
Degree $2$
Conductor $966$
Sign $0.693 + 0.720i$
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  + (−0.415 − 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (1.75 + 1.12i)5-s + (−0.654 − 0.755i)6-s + (0.142 + 0.989i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (0.296 − 2.06i)10-s + (1.76 − 3.87i)11-s + (−0.415 + 0.909i)12-s + (0.330 − 2.29i)13-s + (0.841 − 0.540i)14-s + (1.99 + 0.587i)15-s + (−0.142 − 0.989i)16-s + (−0.299 − 0.345i)17-s + ⋯
L(s)  = 1  + (−0.293 − 0.643i)2-s + (0.553 − 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.784 + 0.503i)5-s + (−0.267 − 0.308i)6-s + (0.0537 + 0.374i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.0938 − 0.652i)10-s + (0.533 − 1.16i)11-s + (−0.119 + 0.262i)12-s + (0.0916 − 0.637i)13-s + (0.224 − 0.144i)14-s + (0.516 + 0.151i)15-s + (−0.0355 − 0.247i)16-s + (−0.0726 − 0.0838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.693 + 0.720i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.693 + 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78576 - 0.759609i\)
\(L(\frac12)\) \(\approx\) \(1.78576 - 0.759609i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (-0.959 + 0.281i)T \)
7 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-4.22 - 2.27i)T \)
good5 \( 1 + (-1.75 - 1.12i)T + (2.07 + 4.54i)T^{2} \)
11 \( 1 + (-1.76 + 3.87i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.330 + 2.29i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (0.299 + 0.345i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (2.93 - 3.38i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (1.60 + 1.84i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-8.16 - 2.39i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-5.93 + 3.81i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-9.64 - 6.19i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-6.60 + 1.93i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 5.22T + 47T^{2} \)
53 \( 1 + (-1.99 - 13.9i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-1.90 + 13.2i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (11.7 + 3.45i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (6.47 + 14.1i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (-3.83 - 8.39i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (8.82 - 10.1i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-0.431 + 3.00i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (9.76 - 6.27i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-2.01 + 0.590i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (6.89 + 4.42i)T + (40.2 + 88.2i)T^{2} \)
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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

−9.820695636641223380931105223245, −9.204206831632990097199641469601, −8.385080528055780032418323537015, −7.68029684008890658635915425178, −6.34994866574559695600623382137, −5.80787903360657037370443866050, −4.33133641325699631217838245624, −3.16328423685438172614228690826, −2.49772185988045402157871623956, −1.17600651019158007280569281694, 1.32175781796217619275931796610, 2.49778923629658877338897622466, 4.26685783720171742923673955314, 4.70190852275952502847125778710, 5.98205260233040051327228166362, 6.84832157812277713694759622514, 7.53697473091055874757796561655, 8.674664173188553203452743628828, 9.203980190827174992076287618991, 9.802912018346394342060037975097

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