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Properties

Label 2-1148-41.31-c1-0-5
Degree $2$
Conductor $1148$
Sign $-0.977 - 0.209i$
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  + 2.28i·3-s + (1.15 + 3.54i)5-s + (0.587 + 0.809i)7-s − 2.21·9-s + (−0.170 − 0.0554i)11-s + (−1.72 + 2.38i)13-s + (−8.10 + 2.63i)15-s + (2.20 + 0.714i)17-s + (−0.238 − 0.328i)19-s + (−1.84 + 1.34i)21-s + (0.682 + 0.495i)23-s + (−7.21 + 5.24i)25-s + 1.78i·27-s + (4.71 − 1.53i)29-s + (1.43 − 4.41i)31-s + ⋯
L(s)  = 1  + 1.31i·3-s + (0.515 + 1.58i)5-s + (0.222 + 0.305i)7-s − 0.739·9-s + (−0.0514 − 0.0167i)11-s + (−0.479 + 0.660i)13-s + (−2.09 + 0.680i)15-s + (0.533 + 0.173i)17-s + (−0.0547 − 0.0754i)19-s + (−0.403 + 0.293i)21-s + (0.142 + 0.103i)23-s + (−1.44 + 1.04i)25-s + 0.343i·27-s + (0.875 − 0.284i)29-s + (0.257 − 0.792i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.977 - 0.209i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.977 - 0.209i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.731278680\)
\(L(\frac12)\) \(\approx\) \(1.731278680\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 + 6.15i)T \)
good3 \( 1 - 2.28iT - 3T^{2} \)
5 \( 1 + (-1.15 - 3.54i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (0.170 + 0.0554i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.72 - 2.38i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.20 - 0.714i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.238 + 0.328i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.682 - 0.495i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-4.71 + 1.53i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.43 + 4.41i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.20 - 3.70i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (0.780 + 0.567i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-7.77 + 10.6i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (10.8 - 3.52i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.25 + 0.915i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.80 + 1.30i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.84 + 1.24i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.775 + 0.252i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 8.02iT - 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + (5.60 + 7.70i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.78 - 0.581i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.10263321283392847650437797103, −9.693181356333300245446992975164, −8.808270606758233887508884718799, −7.64689165660792272251264160843, −6.77081288939366994333927241096, −5.93913275318447153819997464021, −5.00445009663259371152112593127, −4.00834863521938134775511198119, −3.10130065065633854115313858930, −2.17184823623812288787926309025, 0.800314447000585794719095307501, 1.51093686502476970131252221469, 2.75691146121530462348020857744, 4.41453219067177193626598311830, 5.21345827795797839019351785840, 6.03089764790546497379772260515, 7.00405761512083152139168484942, 7.932149606943597234846480583500, 8.325681771059094136646470026107, 9.343484255030301562471453701051

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