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Properties

Label 2-1148-41.18-c1-0-4
Degree $2$
Conductor $1148$
Sign $-0.999 - 0.0269i$
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  − 2.86·3-s + (−3.47 + 2.52i)5-s + (0.309 − 0.951i)7-s + 5.18·9-s + (1.93 + 1.40i)11-s + (0.878 + 2.70i)13-s + (9.93 − 7.22i)15-s + (6.32 + 4.59i)17-s + (−0.619 + 1.90i)19-s + (−0.884 + 2.72i)21-s + (0.422 + 1.30i)23-s + (4.15 − 12.7i)25-s − 6.24·27-s + (−3.17 + 2.30i)29-s + (−2.30 − 1.67i)31-s + ⋯
L(s)  = 1  − 1.65·3-s + (−1.55 + 1.12i)5-s + (0.116 − 0.359i)7-s + 1.72·9-s + (0.582 + 0.422i)11-s + (0.243 + 0.750i)13-s + (2.56 − 1.86i)15-s + (1.53 + 1.11i)17-s + (−0.142 + 0.437i)19-s + (−0.192 + 0.593i)21-s + (0.0881 + 0.271i)23-s + (0.830 − 2.55i)25-s − 1.20·27-s + (−0.590 + 0.428i)29-s + (−0.413 − 0.300i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.999 - 0.0269i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.999 - 0.0269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3683314769\)
\(L(\frac12)\) \(\approx\) \(0.3683314769\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-1.20 - 6.28i)T \)
good3 \( 1 + 2.86T + 3T^{2} \)
5 \( 1 + (3.47 - 2.52i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-1.93 - 1.40i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.878 - 2.70i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-6.32 - 4.59i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.619 - 1.90i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.422 - 1.30i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.17 - 2.30i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.30 + 1.67i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (9.27 - 6.73i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (0.353 + 1.08i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-0.654 - 2.01i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.40 + 3.20i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.97 + 9.14i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.65 - 5.07i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.74 + 6.35i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-5.04 - 3.66i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 9.96T + 73T^{2} \)
79 \( 1 - 8.99T + 79T^{2} \)
83 \( 1 + 3.51T + 83T^{2} \)
89 \( 1 + (-1.60 + 4.92i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (11.6 - 8.49i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.56334729726730365181534068818, −9.779130809062178159713370364860, −8.271346735563155999866170046501, −7.49013129783147843540063214481, −6.79122750089192629020957507578, −6.22779709774727671596035317196, −5.08909060283108501429095433067, −4.03211873891054560847591349123, −3.53349194478843617145043890942, −1.39200498580718646967584417510, 0.27252430473050798791894323767, 1.07684102148332445497606718850, 3.44364193184870277947290283075, 4.33088215146066342732498899949, 5.38355954923778578931584055209, 5.52535746399286675590998750055, 6.98095249343186226744510026742, 7.60912061458062141184906200912, 8.560746400499156036671695161374, 9.330354794632231062457994476490

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