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Properties

Label 2-1148-41.32-c1-0-2
Degree $2$
Conductor $1148$
Sign $-0.980 - 0.196i$
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  + (−2.33 + 2.33i)3-s − 3.90i·5-s + (−0.707 + 0.707i)7-s − 7.90i·9-s + (0.604 − 0.604i)11-s + (0.252 − 0.252i)13-s + (9.11 + 9.11i)15-s + (2.82 + 2.82i)17-s + (−1.41 − 1.41i)19-s − 3.30i·21-s − 8.08·23-s − 10.2·25-s + (11.4 + 11.4i)27-s + (−5.73 + 5.73i)29-s + 10.8·31-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)3-s − 1.74i·5-s + (−0.267 + 0.267i)7-s − 2.63i·9-s + (0.182 − 0.182i)11-s + (0.0699 − 0.0699i)13-s + (2.35 + 2.35i)15-s + (0.685 + 0.685i)17-s + (−0.324 − 0.324i)19-s − 0.720i·21-s − 1.68·23-s − 2.05·25-s + (2.20 + 2.20i)27-s + (−1.06 + 1.06i)29-s + 1.95·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.980 - 0.196i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1468168496\)
\(L(\frac12)\) \(\approx\) \(0.1468168496\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (5.11 + 3.85i)T \)
good3 \( 1 + (2.33 - 2.33i)T - 3iT^{2} \)
5 \( 1 + 3.90iT - 5T^{2} \)
11 \( 1 + (-0.604 + 0.604i)T - 11iT^{2} \)
13 \( 1 + (-0.252 + 0.252i)T - 13iT^{2} \)
17 \( 1 + (-2.82 - 2.82i)T + 17iT^{2} \)
19 \( 1 + (1.41 + 1.41i)T + 19iT^{2} \)
23 \( 1 + 8.08T + 23T^{2} \)
29 \( 1 + (5.73 - 5.73i)T - 29iT^{2} \)
31 \( 1 - 10.8T + 31T^{2} \)
37 \( 1 - 0.470T + 37T^{2} \)
43 \( 1 - 11.3iT - 43T^{2} \)
47 \( 1 + (-1.33 - 1.33i)T + 47iT^{2} \)
53 \( 1 + (4.21 - 4.21i)T - 53iT^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 8.63iT - 61T^{2} \)
67 \( 1 + (5.76 + 5.76i)T + 67iT^{2} \)
71 \( 1 + (-1.36 + 1.36i)T - 71iT^{2} \)
73 \( 1 - 1.88iT - 73T^{2} \)
79 \( 1 + (5.15 - 5.15i)T - 79iT^{2} \)
83 \( 1 + 8.02T + 83T^{2} \)
89 \( 1 + (-1.89 + 1.89i)T - 89iT^{2} \)
97 \( 1 + (-3.77 - 3.77i)T + 97iT^{2} \)
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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

−10.04572370135041747833037323572, −9.534461046656460729934568238417, −8.807489068060820822471088721638, −7.996691794873176906914168320475, −6.27579973560718039761563208104, −5.85232211701505210214126532800, −4.96239644058778398360611103064, −4.41036416407047308359028136437, −3.53456405037474863091729035217, −1.27108810112906479403305949234, 0.082216998920865450285159951108, 1.79261603204937095101369988330, 2.79787343722221079528655679440, 4.17651180444333426517223204168, 5.63399067936492598344240661661, 6.22487262253225599660332695258, 6.80633118727662505539435826925, 7.49432677058652523627958066828, 8.071733797334061622808830803907, 9.960452117500291256833784119521

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