Properties

Label 2-1148-1148.1075-c0-0-1
Degree $2$
Conductor $1148$
Sign $-0.489 + 0.872i$
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  + (−0.866 + 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (0.707 + 0.707i)6-s + (−0.965 − 0.258i)7-s + 0.999i·8-s + (−0.499 + 0.866i)10-s + (−0.258 − 0.965i)11-s + (−0.965 − 0.258i)12-s + (−1 − i)13-s + (0.965 − 0.258i)14-s + (−0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)19-s − 0.999i·20-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.258 − 0.965i)3-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (0.707 + 0.707i)6-s + (−0.965 − 0.258i)7-s + 0.999i·8-s + (−0.499 + 0.866i)10-s + (−0.258 − 0.965i)11-s + (−0.965 − 0.258i)12-s + (−1 − i)13-s + (0.965 − 0.258i)14-s + (−0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)19-s − 0.999i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.489 + 0.872i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ -0.489 + 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5594478332\)
\(L(\frac12)\) \(\approx\) \(0.5594478332\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + iT \)
good3 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.93 + 0.517i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.811404883985172018493024071428, −8.873510024292664391647033234711, −7.959652056200856276920188754452, −7.33381311248500230038842921724, −6.28273542905788947492743258069, −5.99360730750762572113802779078, −5.01679698314042166956293143654, −3.13653674849270993982662471093, −1.88651350847189259923259994460, −0.65460700553976964454035647515, 2.09912607309863675250599161934, 2.79990947383315099633901214668, 4.13558583923746475716291324653, 4.95946552049353758020768421571, 6.39915769173234334043218242052, 6.85661240686749591606183219748, 7.896758428616271583599478098406, 9.329397267825249841931657802016, 9.534613674452790166954385904040, 10.03402061480628518162239198459

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