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Properties

Label 2-1148-287.163-c1-0-6
Degree $2$
Conductor $1148$
Sign $0.0890 - 0.996i$
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  + (0.507 − 0.293i)3-s + (1.03 − 1.79i)5-s + (−1.44 + 2.21i)7-s + (−1.32 + 2.30i)9-s + (−0.222 + 0.128i)11-s + 3.45i·13-s − 1.21i·15-s + (−4.04 + 2.33i)17-s + (1.80 + 1.04i)19-s + (−0.0812 + 1.54i)21-s + (−1.62 + 2.81i)23-s + (0.347 + 0.602i)25-s + 3.31i·27-s + 3.70i·29-s + (−2.78 − 4.82i)31-s + ⋯
L(s)  = 1  + (0.293 − 0.169i)3-s + (0.463 − 0.803i)5-s + (−0.544 + 0.838i)7-s + (−0.442 + 0.766i)9-s + (−0.0671 + 0.0387i)11-s + 0.958i·13-s − 0.314i·15-s + (−0.981 + 0.566i)17-s + (0.413 + 0.238i)19-s + (−0.0177 + 0.338i)21-s + (−0.339 + 0.587i)23-s + (0.0695 + 0.120i)25-s + 0.638i·27-s + 0.688i·29-s + (−0.499 − 0.865i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0890 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0890 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.0890 - 0.996i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.0890 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.308185153\)
\(L(\frac12)\) \(\approx\) \(1.308185153\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 - 2.21i)T \)
41 \( 1 + (6.37 + 0.640i)T \)
good3 \( 1 + (-0.507 + 0.293i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.03 + 1.79i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.222 - 0.128i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.45iT - 13T^{2} \)
17 \( 1 + (4.04 - 2.33i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.80 - 1.04i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.62 - 2.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.70iT - 29T^{2} \)
31 \( 1 + (2.78 + 4.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.268 - 0.465i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 8.37T + 43T^{2} \)
47 \( 1 + (-7.09 - 4.09i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.20 - 3.00i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.959 - 1.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.28 + 7.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.10 + 4.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.57iT - 71T^{2} \)
73 \( 1 + (-1.75 - 3.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.43 - 4.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 + (6.91 + 3.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.39iT - 97T^{2} \)
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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

−9.684714814054271569871293764640, −9.105886290053387641795983765940, −8.582333865909946408095863981483, −7.63772448855290685849645580530, −6.57803202823656025124912748963, −5.68263461893985559003330343179, −5.01874356894426496092152838738, −3.84162436973086911632187421899, −2.50681572385946876284646879958, −1.69873825494148036098679835381, 0.52382061884308391102812986382, 2.49658979064592258160810823910, 3.23265933061198012661827592881, 4.17199252161227923977766372288, 5.45009624007196878413287963755, 6.45604272723636358567688816747, 6.92751079393659959237240910912, 7.947738690971457537558155133235, 8.932278224903127799102605652851, 9.671492001203010682803926905579

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