Properties

Label 2-966-161.160-c1-0-13
Degree $2$
Conductor $966$
Sign $0.551 - 0.834i$
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  − 2-s + i·3-s + 4-s + 2.64·5-s i·6-s + 2.64·7-s − 8-s − 9-s − 2.64·10-s + i·12-s + 5i·13-s − 2.64·14-s + 2.64i·15-s + 16-s + 5.29·17-s + 18-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s + 1.18·5-s − 0.408i·6-s + 0.999·7-s − 0.353·8-s − 0.333·9-s − 0.836·10-s + 0.288i·12-s + 1.38i·13-s − 0.707·14-s + 0.683i·15-s + 0.250·16-s + 1.28·17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.551 - 0.834i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.551 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39286 + 0.748693i\)
\(L(\frac12)\) \(\approx\) \(1.39286 + 0.748693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - iT \)
7 \( 1 - 2.64T \)
23 \( 1 + (-4 - 2.64i)T \)
good5 \( 1 - 2.64T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 7.93iT - 37T^{2} \)
41 \( 1 + 9iT - 41T^{2} \)
43 \( 1 + 2.64iT - 43T^{2} \)
47 \( 1 + 13iT - 47T^{2} \)
53 \( 1 - 5.29iT - 53T^{2} \)
59 \( 1 - 14iT - 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 5.29iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 5.29iT - 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 2.64T + 97T^{2} \)
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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

−10.14444912949362279440458366985, −9.247682139922560713397118727930, −8.801731506877256099932448124253, −7.78380790687157040839730208143, −6.78550490346455012978291349800, −5.82556518616959511608893785153, −5.04292540786446870747672661698, −3.90303665285404547679457444820, −2.36113565905583445907627168562, −1.49666236098170502977052883662, 1.04196136999672848001713736809, 2.03490491729466861780872188586, 3.08272869373352600458425512355, 4.90958791956685076774624614007, 5.75055526459804265711128335384, 6.45804986666074199733877901601, 7.65731809059400648087871537302, 8.112368121385794831192058932784, 9.019965241485599254623004252709, 9.891938190476326948446195388047

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