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Properties

Label 2-1169-1169.667-c0-0-8
Degree $2$
Conductor $1169$
Sign $-0.401 + 0.916i$
Analytic cond. $0.583406$
Root an. cond. $0.763810$
Motivic weight $0$
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 + 0.719i)2-s + (−0.928 − 1.60i)3-s + (0.154 + 0.268i)4-s + 1.54·6-s + (0.0475 + 0.998i)7-s − 1.08·8-s + (−1.22 + 2.11i)9-s + (−0.981 − 1.70i)11-s + (0.287 − 0.498i)12-s + (−0.738 − 0.380i)14-s + (0.297 − 0.514i)16-s + (−1.01 − 1.76i)18-s + (0.959 − 1.66i)19-s + (1.56 − 1.00i)21-s + 1.63·22-s + ⋯
L(s)  = 1  + (−0.415 + 0.719i)2-s + (−0.928 − 1.60i)3-s + (0.154 + 0.268i)4-s + 1.54·6-s + (0.0475 + 0.998i)7-s − 1.08·8-s + (−1.22 + 2.11i)9-s + (−0.981 − 1.70i)11-s + (0.287 − 0.498i)12-s + (−0.738 − 0.380i)14-s + (0.297 − 0.514i)16-s + (−1.01 − 1.76i)18-s + (0.959 − 1.66i)19-s + (1.56 − 1.00i)21-s + 1.63·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1169\)    =    \(7 \cdot 167\)
Sign: $-0.401 + 0.916i$
Analytic conductor: \(0.583406\)
Root analytic conductor: \(0.763810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1169} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1169,\ (\ :0),\ -0.401 + 0.916i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3244966832\)
\(L(\frac12)\) \(\approx\) \(0.3244966832\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.0475 - 0.998i)T \)
167 \( 1 - T \)
good2 \( 1 + (0.415 - 0.719i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.959 + 1.66i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.77T + T^{2} \)
31 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.142 + 0.246i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.786 + 1.36i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.30T + T^{2} \)
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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

−9.333725387572172602234339891934, −8.532062322064126977218155088739, −7.87167844827223407680215893157, −7.32989974979234884188494523814, −6.37340228517651572558674005032, −5.76825870827317733876703672809, −5.30219650768926865360324625191, −3.05654130876617392438223044095, −2.25601632833754889156856513200, −0.35010044936447848775972091567, 1.67193666376029192835402814649, 3.34991354213440264050755644685, 4.06736351728471161814699188074, 5.20111400371520974163517504040, 5.60231213454831174567206742652, 6.88458466874052251302365300498, 7.80296735449342793673497681499, 9.328649370002164948545601000538, 9.703803181066175017479040027190, 10.32772112135210757907279393773

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