Properties

Label 2-1148-41.18-c1-0-3
Degree $2$
Conductor $1148$
Sign $-0.393 - 0.919i$
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  − 2.49·3-s + (0.335 − 0.243i)5-s + (0.309 − 0.951i)7-s + 3.21·9-s + (−2.96 − 2.15i)11-s + (1.82 + 5.61i)13-s + (−0.835 + 0.606i)15-s + (−2.74 − 1.99i)17-s + (0.896 − 2.75i)19-s + (−0.770 + 2.37i)21-s + (−1.87 − 5.76i)23-s + (−1.49 + 4.59i)25-s − 0.524·27-s + (4.51 − 3.27i)29-s + (1.80 + 1.31i)31-s + ⋯
L(s)  = 1  − 1.43·3-s + (0.149 − 0.108i)5-s + (0.116 − 0.359i)7-s + 1.07·9-s + (−0.895 − 0.650i)11-s + (0.506 + 1.55i)13-s + (−0.215 + 0.156i)15-s + (−0.666 − 0.483i)17-s + (0.205 − 0.632i)19-s + (−0.168 + 0.517i)21-s + (−0.390 − 1.20i)23-s + (−0.298 + 0.918i)25-s − 0.100·27-s + (0.837 − 0.608i)29-s + (0.324 + 0.235i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.393 - 0.919i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.393 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3883535864\)
\(L(\frac12)\) \(\approx\) \(0.3883535864\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.21 - 3.71i)T \)
good3 \( 1 + 2.49T + 3T^{2} \)
5 \( 1 + (-0.335 + 0.243i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (2.96 + 2.15i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.82 - 5.61i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.74 + 1.99i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.896 + 2.75i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.87 + 5.76i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.51 + 3.27i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.80 - 1.31i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.56 - 1.86i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-2.93 - 9.03i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-2.29 - 7.07i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.86 - 5.71i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.0716 + 0.220i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.19 - 9.82i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.04 - 6.56i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (3.21 + 2.33i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 2.86T + 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + (2.10 - 6.48i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-11.7 + 8.51i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.32935200077460389979528630765, −9.310178362470632096642261652083, −8.479594261935599814190221885346, −7.36116669122793481084091819486, −6.46887433200620118880150883661, −6.00680386547698052494802148109, −4.82254196931541173251276797138, −4.41698019792117020230229668249, −2.77611811465413412057021161114, −1.20262544711054714880500540580, 0.22472562309755801783440905706, 1.87108249982408437847126942528, 3.30411039497061310453482974413, 4.63602829508312654540970162405, 5.50992168552351805718502133701, 5.87608296872573120054545425928, 6.88790046114137833805497136124, 7.85280711828038333961589406370, 8.609577474797080818277606013632, 10.00091799856893566547741131054

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