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Properties

Label 2-1148-287.163-c1-0-0
Degree $2$
Conductor $1148$
Sign $0.303 + 0.952i$
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  + (−1.90 + 1.09i)3-s + (−1.32 + 2.30i)5-s + (−2.55 + 0.693i)7-s + (0.913 − 1.58i)9-s + (−4.86 + 2.81i)11-s + 0.607i·13-s − 5.83i·15-s + (1.42 − 0.820i)17-s + (−1.82 − 1.05i)19-s + (4.09 − 4.12i)21-s + (−3.27 + 5.66i)23-s + (−1.02 − 1.78i)25-s − 2.57i·27-s + 8.24i·29-s + (−0.755 − 1.30i)31-s + ⋯
L(s)  = 1  + (−1.09 + 0.634i)3-s + (−0.594 + 1.02i)5-s + (−0.965 + 0.261i)7-s + (0.304 − 0.527i)9-s + (−1.46 + 0.847i)11-s + 0.168i·13-s − 1.50i·15-s + (0.344 − 0.199i)17-s + (−0.419 − 0.242i)19-s + (0.894 − 0.899i)21-s + (−0.681 + 1.18i)23-s + (−0.205 − 0.356i)25-s − 0.495i·27-s + 1.53i·29-s + (−0.135 − 0.234i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.303 + 0.952i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.303 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09816346762\)
\(L(\frac12)\) \(\approx\) \(0.09816346762\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.55 - 0.693i)T \)
41 \( 1 + (-3.30 + 5.48i)T \)
good3 \( 1 + (1.90 - 1.09i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.32 - 2.30i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.86 - 2.81i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.607iT - 13T^{2} \)
17 \( 1 + (-1.42 + 0.820i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.82 + 1.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.27 - 5.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.24iT - 29T^{2} \)
31 \( 1 + (0.755 + 1.30i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.09 + 1.89i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 9.42T + 43T^{2} \)
47 \( 1 + (-3.70 - 2.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.13 - 2.38i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.353 + 0.612i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 - 5.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.79 - 5.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.07iT - 71T^{2} \)
73 \( 1 + (5.47 + 9.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.65 - 2.68i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 + (-4.05 - 2.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.0178iT - 97T^{2} \)
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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

−10.65601210682879501758553112916, −9.908231041241358674626690565604, −9.081473595985867461146110474798, −7.60186554874872052181470463872, −7.23412928787801672490022022265, −6.10466643062505481261372684090, −5.47616968574531389241006003093, −4.49021307207749388949810090409, −3.43342663621709478260518550986, −2.46931343141083422537490827247, 0.07153204679851462647496053700, 0.78829913172933707088816739733, 2.66349518245293834029120896504, 3.97802791135340488520061239072, 4.96556900654045868875341717273, 5.93024387258410104292546617471, 6.32206670927253597061116520419, 7.62205375804820605022394094665, 8.115120086434396923479490465067, 9.064575994203978031365176789317

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