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Properties

Label 2-1148-41.37-c1-0-8
Degree $2$
Conductor $1148$
Sign $0.556 + 0.831i$
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  − 3.33·3-s + (−0.141 + 0.435i)5-s + (−0.809 − 0.587i)7-s + 8.11·9-s + (−0.348 − 1.07i)11-s + (−5.36 + 3.90i)13-s + (0.471 − 1.45i)15-s + (0.0378 + 0.116i)17-s + (−1.20 − 0.876i)19-s + (2.69 + 1.95i)21-s + (−1.81 + 1.31i)23-s + (3.87 + 2.81i)25-s − 17.0·27-s + (−1.94 + 5.97i)29-s + (−0.240 − 0.738i)31-s + ⋯
L(s)  = 1  − 1.92·3-s + (−0.0632 + 0.194i)5-s + (−0.305 − 0.222i)7-s + 2.70·9-s + (−0.104 − 0.323i)11-s + (−1.48 + 1.08i)13-s + (0.121 − 0.374i)15-s + (0.00917 + 0.0282i)17-s + (−0.276 − 0.201i)19-s + (0.588 + 0.427i)21-s + (−0.378 + 0.275i)23-s + (0.775 + 0.563i)25-s − 3.28·27-s + (−0.360 + 1.10i)29-s + (−0.0431 − 0.132i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.556 + 0.831i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.556 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4699188234\)
\(L(\frac12)\) \(\approx\) \(0.4699188234\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-3.79 + 5.15i)T \)
good3 \( 1 + 3.33T + 3T^{2} \)
5 \( 1 + (0.141 - 0.435i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (0.348 + 1.07i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (5.36 - 3.90i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.0378 - 0.116i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.20 + 0.876i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.81 - 1.31i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.94 - 5.97i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.240 + 0.738i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.08 + 6.40i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (0.429 - 0.311i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-0.0152 + 0.0110i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.48 + 4.55i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.77 + 7.10i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (6.92 + 5.03i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.98 + 9.17i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.32 + 10.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 - 7.92T + 73T^{2} \)
79 \( 1 - 6.89T + 79T^{2} \)
83 \( 1 - 4.78T + 83T^{2} \)
89 \( 1 + (-12.8 - 9.31i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.06 + 6.35i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.798420821225651001574487988094, −9.224677121962123695890313574768, −7.58962601518923217945085231894, −6.97074239651664314385120808801, −6.37319621208090264014735873522, −5.35046255665624955731983753501, −4.77073658614197063117824408321, −3.74754266638261068495623056342, −1.96707995654338465072997960231, −0.38350721626123447050085661691, 0.815187790133706513197787581326, 2.50480419357449728786454411341, 4.22683468087767386994679559004, 4.94107593735969704775347998410, 5.66589313829102951768918704393, 6.42102673490886591932059370278, 7.25282208641473678277601727111, 8.050655865305688728169111811463, 9.509818550828244455678046804502, 10.16543466339905873761720908338

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