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Properties

Label 2-1148-287.163-c1-0-11
Degree $2$
Conductor $1148$
Sign $0.781 - 0.623i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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  + (0.900 − 0.520i)3-s + (−0.292 + 0.506i)5-s + (2.31 − 1.27i)7-s + (−0.958 + 1.66i)9-s + (−3.20 + 1.84i)11-s + 5.26i·13-s + 0.608i·15-s + (6.11 − 3.53i)17-s + (1.31 + 0.756i)19-s + (1.42 − 2.35i)21-s + (−0.874 + 1.51i)23-s + (2.32 + 4.03i)25-s + 5.11i·27-s + 5.75i·29-s + (−0.171 − 0.297i)31-s + ⋯
L(s)  = 1  + (0.520 − 0.300i)3-s + (−0.130 + 0.226i)5-s + (0.875 − 0.483i)7-s + (−0.319 + 0.553i)9-s + (−0.965 + 0.557i)11-s + 1.46i·13-s + 0.157i·15-s + (1.48 − 0.856i)17-s + (0.300 + 0.173i)19-s + (0.310 − 0.514i)21-s + (−0.182 + 0.315i)23-s + (0.465 + 0.806i)25-s + 0.984i·27-s + 1.06i·29-s + (−0.0308 − 0.0534i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.781 - 0.623i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.934432395\)
\(L(\frac12)\) \(\approx\) \(1.934432395\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.31 + 1.27i)T \)
41 \( 1 + (-5.85 - 2.59i)T \)
good3 \( 1 + (-0.900 + 0.520i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.292 - 0.506i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.20 - 1.84i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.26iT - 13T^{2} \)
17 \( 1 + (-6.11 + 3.53i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.31 - 0.756i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.874 - 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.75iT - 29T^{2} \)
31 \( 1 + (0.171 + 0.297i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.45 + 9.45i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 6.68T + 43T^{2} \)
47 \( 1 + (-3.16 - 1.82i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.70 - 3.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.32 + 7.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.89 + 8.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.77 + 3.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.578iT - 71T^{2} \)
73 \( 1 + (-1.30 - 2.26i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.98 - 4.03i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + (-5.86 - 3.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.7iT - 97T^{2} \)
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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

−9.768930543683903490295023753153, −9.099492826500638103120101204555, −7.901276013084596478515151178564, −7.65017141888593318691885173213, −6.88925870067227221906386867140, −5.40064063253109971204740369420, −4.84624561961737872955668590202, −3.61566775670771359859926961823, −2.49936202460085720092785027807, −1.47875132432450888295738718177, 0.862913921917186336298596869609, 2.61185445238452421361296819039, 3.30122861697597554760039679494, 4.51414567713443647005749346773, 5.54546094840412695401858812272, 6.03000106659114987028231700485, 7.66641572455022230515523593823, 8.252928832411146178626255670354, 8.546275056096450449688708559252, 9.853130701215453655029833772419

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