Properties

Label 2-1148-1148.163-c0-0-7
Degree $2$
Conductor $1148$
Sign $-0.750 - 0.660i$
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 + (−0.965 − 1.67i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s − 1.93·6-s + (0.258 + 0.965i)7-s − 0.999·8-s + (−1.36 + 2.36i)9-s + (−0.866 − 1.5i)10-s + (−0.258 − 0.448i)11-s + (−0.965 + 1.67i)12-s + (0.965 + 0.258i)14-s − 3.34·15-s + (−0.5 + 0.866i)16-s + (1.36 + 2.36i)18-s + (0.258 − 0.448i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.965 − 1.67i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s − 1.93·6-s + (0.258 + 0.965i)7-s − 0.999·8-s + (−1.36 + 2.36i)9-s + (−0.866 − 1.5i)10-s + (−0.258 − 0.448i)11-s + (−0.965 + 1.67i)12-s + (0.965 + 0.258i)14-s − 3.34·15-s + (−0.5 + 0.866i)16-s + (1.36 + 2.36i)18-s + (0.258 − 0.448i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.750 - 0.660i$
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: \((2,\ 1148,\ (\ :0),\ -0.750 - 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9728526455\)
\(L(\frac12)\) \(\approx\) \(0.9728526455\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
41 \( 1 - T \)
good3 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.93T + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.440653766532190890426792394619, −8.714440068813316676325374459663, −8.020282238570439747623032287164, −6.63272392355241581661089153855, −5.65129426239334469057065288301, −5.53228506996965252126231164317, −4.66694523841506889504044113411, −2.58665123259185702685698889829, −1.81937357046570986560164506648, −0.866174618577696087774110532668, 2.91138898923709232084090716102, 3.79541892039467770517004050140, 4.56650684387378521796771055308, 5.49748251868575040280317254808, 6.15951033267591336138433258016, 6.86619554014286051767549694693, 7.75753123428272403574264355514, 9.198203905077416404482021584055, 9.829715363034550421037420676803, 10.47094033174420814752691764867

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