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Properties

Label 2-1148-41.18-c1-0-0
Degree $2$
Conductor $1148$
Sign $-0.881 - 0.471i$
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  − 0.414·3-s + (0.335 − 0.243i)5-s + (0.309 − 0.951i)7-s − 2.82·9-s + (−1.5 − 1.08i)11-s + (−0.286 − 0.880i)13-s + (−0.138 + 0.100i)15-s + (−1.49 − 1.08i)17-s + (−1.82 + 5.61i)19-s + (−0.127 + 0.393i)21-s + (1.27 + 3.92i)23-s + (−1.49 + 4.59i)25-s + 2.41·27-s + (−3.39 + 2.46i)29-s + (−4.11 − 2.99i)31-s + ⋯
L(s)  = 1  − 0.239·3-s + (0.149 − 0.108i)5-s + (0.116 − 0.359i)7-s − 0.942·9-s + (−0.452 − 0.328i)11-s + (−0.0793 − 0.244i)13-s + (−0.0358 + 0.0260i)15-s + (−0.362 − 0.263i)17-s + (−0.418 + 1.28i)19-s + (−0.0279 + 0.0859i)21-s + (0.266 + 0.819i)23-s + (−0.298 + 0.918i)25-s + 0.464·27-s + (−0.630 + 0.457i)29-s + (−0.739 − 0.537i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.881 - 0.471i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2175643201\)
\(L(\frac12)\) \(\approx\) \(0.2175643201\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.74 - 6.16i)T \)
good3 \( 1 + 0.414T + 3T^{2} \)
5 \( 1 + (-0.335 + 0.243i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (1.5 + 1.08i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.286 + 0.880i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.49 + 1.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.82 - 5.61i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.27 - 3.92i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.39 - 2.46i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.11 + 2.99i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.56 - 1.86i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (1.97 + 6.07i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (1.67 + 5.16i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.26 + 3.82i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.747 + 2.29i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.64 - 11.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.68 - 7.03i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (5.93 + 4.31i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 0.191T + 73T^{2} \)
79 \( 1 - 7.16T + 79T^{2} \)
83 \( 1 + 0.886T + 83T^{2} \)
89 \( 1 + (5.55 - 17.1i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (15.7 - 11.4i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.22934886869021930597903447502, −9.309948012465725299041270394268, −8.486547866989016499911952195886, −7.73744131182550365119049090348, −6.83878793355761585374439322439, −5.66037627334592500510877200187, −5.35255874894346602681881208807, −3.96318858979800630436361709099, −3.03290459080168996623979628637, −1.66064805872088054091065391080, 0.090933347684598380580040214039, 2.10180884816231787293277401269, 2.94171140771920665512070621963, 4.36423307065726066305201996924, 5.18497118915964653764213888370, 6.09355729694283347685626530003, 6.83280466543950439636196657761, 7.87481445354871625854450185695, 8.757425530976711442520864262411, 9.280450635860735933494258516803

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