Properties

Label 2-1148-1148.163-c0-0-3
Degree $2$
Conductor $1148$
Sign $-0.0633 - 0.997i$
Analytic cond. $0.572926$
Root an. cond. $0.756919$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.499 − 0.866i)14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)19-s − 0.999·20-s − 0.999·21-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s − 0.999·6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (0.499 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.499 − 0.866i)14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)19-s − 0.999·20-s − 0.999·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001938205\)
\(L(\frac12)\) \(\approx\) \(1.001938205\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.700112880000607070960059493996, −9.246254029239740980119232918786, −8.961563577808475987237338630687, −7.955594975165847919820021026106, −6.82588037278540464491630510377, −6.08151520254665653405581156349, −4.97588879065555881578862278760, −4.56925172812211455262890776688, −3.13573954958441692168331243465, −1.56960827996413838761891495272, 1.21075637368525368329866818591, 2.35238718539025944563366193926, 3.26204845692091433435079383733, 4.06596236240117556461458709393, 5.71640639534600054393975751587, 6.87715081433727202897705999294, 7.29191951928437660531015918005, 8.223806072998561826942099750806, 8.986290818922945568309886387979, 10.03449868006302145032090143066

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