Properties

Label 2-966-23.16-c1-0-8
Degree $2$
Conductor $966$
Sign $0.934 - 0.354i$
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  + (−0.654 + 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.544 + 1.19i)5-s + (−0.142 + 0.989i)6-s + (0.959 − 0.281i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−1.25 − 0.368i)10-s + (−2.25 − 2.59i)11-s + (−0.654 − 0.755i)12-s + (3.44 + 1.01i)13-s + (−0.415 + 0.909i)14-s + (1.10 + 0.708i)15-s + (−0.959 + 0.281i)16-s + (0.0988 − 0.687i)17-s + ⋯
L(s)  = 1  + (−0.463 + 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.243 + 0.532i)5-s + (−0.0580 + 0.404i)6-s + (0.362 − 0.106i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.397 − 0.116i)10-s + (−0.679 − 0.783i)11-s + (−0.189 − 0.218i)12-s + (0.956 + 0.280i)13-s + (−0.111 + 0.243i)14-s + (0.284 + 0.182i)15-s + (−0.239 + 0.0704i)16-s + (0.0239 − 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.934 - 0.354i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.934 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62348 + 0.297824i\)
\(L(\frac12)\) \(\approx\) \(1.62348 + 0.297824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 0.755i)T \)
3 \( 1 + (-0.841 + 0.540i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (-4.16 + 2.37i)T \)
good5 \( 1 + (-0.544 - 1.19i)T + (-3.27 + 3.77i)T^{2} \)
11 \( 1 + (2.25 + 2.59i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (-3.44 - 1.01i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.0988 + 0.687i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-1.01 - 7.02i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-1.09 + 7.64i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-5.37 - 3.45i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (2.04 - 4.48i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-0.389 - 0.852i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-1.88 + 1.20i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + 1.20T + 47T^{2} \)
53 \( 1 + (8.09 - 2.37i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-9.77 - 2.86i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (3.42 + 2.19i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (0.204 - 0.235i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (-7.40 + 8.54i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (0.0693 + 0.482i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-15.7 - 4.62i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-0.735 + 1.61i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-3.37 + 2.16i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-3.35 - 7.35i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.09506939144385485349672743079, −9.026236697048459279205671679410, −8.196909467792177866299585937899, −7.84239363300414705574498145982, −6.58689992614385934193478829771, −6.13350159780666478385393131426, −4.97060264204198816735685902846, −3.63930547621306692905022941880, −2.52617784900341705512644257409, −1.12789671601591648521570800613, 1.16689756453629457021908104822, 2.42672955752093120374360822065, 3.42704556898926613902585208107, 4.69879479033475676264189882806, 5.28856126233926352652886078217, 6.82169102856355114861362789688, 7.68729073943582034652219442160, 8.597980365886849678786396736045, 9.106263975671695348499208754340, 9.851223354387912059825008393352

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