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Properties

Label 2-966-161.160-c1-0-14
Degree $2$
Conductor $966$
Sign $0.460 - 0.887i$
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 + 4.19·5-s + i·6-s + (−2.55 + 0.691i)7-s + 8-s − 9-s + 4.19·10-s + 5.99i·11-s + i·12-s − 0.756i·13-s + (−2.55 + 0.691i)14-s + 4.19i·15-s + 16-s − 5.10·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 1.87·5-s + 0.408i·6-s + (−0.965 + 0.261i)7-s + 0.353·8-s − 0.333·9-s + 1.32·10-s + 1.80i·11-s + 0.288i·12-s − 0.209i·13-s + (−0.682 + 0.184i)14-s + 1.08i·15-s + 0.250·16-s − 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.460 - 0.887i$
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: \((2,\ 966,\ (\ :1/2),\ 0.460 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53430 + 1.53978i\)
\(L(\frac12)\) \(\approx\) \(2.53430 + 1.53978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.55 - 0.691i)T \)
23 \( 1 + (-3.53 - 3.24i)T \)
good5 \( 1 - 4.19T + 5T^{2} \)
11 \( 1 - 5.99iT - 11T^{2} \)
13 \( 1 + 0.756iT - 13T^{2} \)
17 \( 1 + 5.10T + 17T^{2} \)
19 \( 1 - 1.02T + 19T^{2} \)
29 \( 1 - 0.951T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 9.30iT - 37T^{2} \)
41 \( 1 + 4.75iT - 41T^{2} \)
43 \( 1 + 7.92iT - 43T^{2} \)
47 \( 1 + 9.62iT - 47T^{2} \)
53 \( 1 - 9.19iT - 53T^{2} \)
59 \( 1 - 0.291iT - 59T^{2} \)
61 \( 1 - 4.08T + 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 + 3.70T + 71T^{2} \)
73 \( 1 + 12.8iT - 73T^{2} \)
79 \( 1 + 3.20iT - 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 1.24T + 89T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−10.26858400230013127003120566944, −9.281785396100929563162060607638, −9.104715955249295309118028382652, −7.17560670449278627287294260896, −6.63395164752437510699617488425, −5.63696897162752952204369680463, −5.10815872801988911792816176232, −4.00047045743817430961754212239, −2.64301233439825789808078496642, −1.97930310972824931392128144116, 1.15616509516127515989703305788, 2.55329248069224456677461320401, 3.17967791083010844769894604866, 4.78484559400770193801574179503, 5.79442136724208117468810351971, 6.45116760548106979329282399078, 6.68793624083298706628636892224, 8.292377561730717591996167919928, 9.118158010079736044095484797046, 9.872783827408010334555764570215

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