Properties

Label 2-1148-41.32-c1-0-8
Degree $2$
Conductor $1148$
Sign $-0.970 - 0.241i$
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  + (−0.868 + 0.868i)3-s + 4.37i·5-s + (−0.707 + 0.707i)7-s + 1.49i·9-s + (1.10 − 1.10i)11-s + (0.155 − 0.155i)13-s + (−3.80 − 3.80i)15-s + (5.27 + 5.27i)17-s + (5.24 + 5.24i)19-s − 1.22i·21-s + 5.28·23-s − 14.1·25-s + (−3.90 − 3.90i)27-s + (−4.12 + 4.12i)29-s + 1.03·31-s + ⋯
L(s)  = 1  + (−0.501 + 0.501i)3-s + 1.95i·5-s + (−0.267 + 0.267i)7-s + 0.497i·9-s + (0.333 − 0.333i)11-s + (0.0432 − 0.0432i)13-s + (−0.982 − 0.982i)15-s + (1.27 + 1.27i)17-s + (1.20 + 1.20i)19-s − 0.268i·21-s + 1.10·23-s − 2.83·25-s + (−0.750 − 0.750i)27-s + (−0.766 + 0.766i)29-s + 0.185·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.970 - 0.241i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.272977085\)
\(L(\frac12)\) \(\approx\) \(1.272977085\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.93 + 4.08i)T \)
good3 \( 1 + (0.868 - 0.868i)T - 3iT^{2} \)
5 \( 1 - 4.37iT - 5T^{2} \)
11 \( 1 + (-1.10 + 1.10i)T - 11iT^{2} \)
13 \( 1 + (-0.155 + 0.155i)T - 13iT^{2} \)
17 \( 1 + (-5.27 - 5.27i)T + 17iT^{2} \)
19 \( 1 + (-5.24 - 5.24i)T + 19iT^{2} \)
23 \( 1 - 5.28T + 23T^{2} \)
29 \( 1 + (4.12 - 4.12i)T - 29iT^{2} \)
31 \( 1 - 1.03T + 31T^{2} \)
37 \( 1 + 2.58T + 37T^{2} \)
43 \( 1 + 8.71iT - 43T^{2} \)
47 \( 1 + (2.00 + 2.00i)T + 47iT^{2} \)
53 \( 1 + (-5.98 + 5.98i)T - 53iT^{2} \)
59 \( 1 - 6.40T + 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 + (1.54 + 1.54i)T + 67iT^{2} \)
71 \( 1 + (-0.123 + 0.123i)T - 71iT^{2} \)
73 \( 1 - 5.51iT - 73T^{2} \)
79 \( 1 + (8.20 - 8.20i)T - 79iT^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + (-11.9 + 11.9i)T - 89iT^{2} \)
97 \( 1 + (-6.20 - 6.20i)T + 97iT^{2} \)
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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

−10.19619777882519983170619623600, −9.852696708685165007771764709602, −8.462454717793880638627043769259, −7.52697798971880548056271037706, −6.86528669934774414945600918797, −5.83531721642809922032905420827, −5.41338883180285608303504023133, −3.62147550318512668391565860459, −3.37126872504452612572573736444, −1.92530820706160333568531353453, 0.68250091907635374994906863816, 1.27629777443485979855889397557, 3.15333440245408392410811330120, 4.43143900952695701409351563665, 5.17275045121145321842174020989, 5.84460054632544109700487981374, 7.07180976070928227715208969804, 7.62844986443073344442481120108, 8.804543774446038968951506593347, 9.399610344860207381906763232306

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