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Properties

Label 2-1148-41.31-c1-0-0
Degree $2$
Conductor $1148$
Sign $-0.658 + 0.752i$
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  + 2.48i·3-s + (−0.290 − 0.894i)5-s + (0.587 + 0.809i)7-s − 3.15·9-s + (−4.02 − 1.30i)11-s + (−0.240 + 0.331i)13-s + (2.21 − 0.721i)15-s + (−5.71 − 1.85i)17-s + (1.00 + 1.38i)19-s + (−2.00 + 1.45i)21-s + (−5.11 − 3.71i)23-s + (3.32 − 2.41i)25-s − 0.390i·27-s + (−9.37 + 3.04i)29-s + (−1.49 + 4.61i)31-s + ⋯
L(s)  = 1  + 1.43i·3-s + (−0.129 − 0.399i)5-s + (0.222 + 0.305i)7-s − 1.05·9-s + (−1.21 − 0.394i)11-s + (−0.0667 + 0.0918i)13-s + (0.572 − 0.186i)15-s + (−1.38 − 0.450i)17-s + (0.230 + 0.317i)19-s + (−0.438 + 0.318i)21-s + (−1.06 − 0.775i)23-s + (0.665 − 0.483i)25-s − 0.0751i·27-s + (−1.74 + 0.565i)29-s + (−0.269 + 0.828i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.658 + 0.752i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.658 + 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1712740697\)
\(L(\frac12)\) \(\approx\) \(0.1712740697\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (4.52 + 4.52i)T \)
good3 \( 1 - 2.48iT - 3T^{2} \)
5 \( 1 + (0.290 + 0.894i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (4.02 + 1.30i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.240 - 0.331i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.71 + 1.85i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.00 - 1.38i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (5.11 + 3.71i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (9.37 - 3.04i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.49 - 4.61i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.163 - 0.504i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-1.57 - 1.14i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-2.18 + 3.00i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.53 - 2.44i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.26 - 4.55i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.42 + 1.03i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.101 - 0.0328i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-10.7 - 3.48i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 12.2iT - 79T^{2} \)
83 \( 1 - 5.83T + 83T^{2} \)
89 \( 1 + (-7.82 - 10.7i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.00 + 1.30i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.47658458228649207931827123832, −9.454082168700797520810466871452, −8.820276801789744532673435582961, −8.176757890442426666507073834340, −7.04349848487421117860332967294, −5.76268397115362367432888782267, −5.04646429021433349021223508066, −4.39853929738798022070573652022, −3.39485601984520666163174754836, −2.23636504657332756246526132658, 0.06819588220161329592963420487, 1.77062190116129595495868205982, 2.51881196340286038689097224513, 3.88847120780208514838601490559, 5.13408197206117501261780368987, 6.11537058677811532583795476583, 6.93471438010490360780790880232, 7.65501338892000955350710806117, 8.021271193126790195526151456831, 9.198353543698695756804200390309

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