Properties

Label 2-1148-41.37-c1-0-5
Degree $2$
Conductor $1148$
Sign $0.984 - 0.174i$
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  − 2.28·3-s + (−0.815 + 2.50i)5-s + (0.809 + 0.587i)7-s + 2.22·9-s + (−1.72 − 5.30i)11-s + (1.00 − 0.727i)13-s + (1.86 − 5.73i)15-s + (−0.144 − 0.443i)17-s + (−2.66 − 1.93i)19-s + (−1.84 − 1.34i)21-s + (0.607 − 0.441i)23-s + (−1.58 − 1.15i)25-s + 1.76·27-s + (−0.433 + 1.33i)29-s + (1.42 + 4.38i)31-s + ⋯
L(s)  = 1  − 1.32·3-s + (−0.364 + 1.12i)5-s + (0.305 + 0.222i)7-s + 0.742·9-s + (−0.519 − 1.60i)11-s + (0.277 − 0.201i)13-s + (0.481 − 1.48i)15-s + (−0.0349 − 0.107i)17-s + (−0.611 − 0.444i)19-s + (−0.403 − 0.293i)21-s + (0.126 − 0.0920i)23-s + (−0.317 − 0.230i)25-s + 0.339·27-s + (−0.0805 + 0.247i)29-s + (0.255 + 0.787i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.984 - 0.174i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.984 - 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8107263217\)
\(L(\frac12)\) \(\approx\) \(0.8107263217\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-3.18 - 5.55i)T \)
good3 \( 1 + 2.28T + 3T^{2} \)
5 \( 1 + (0.815 - 2.50i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (1.72 + 5.30i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.00 + 0.727i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.144 + 0.443i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.66 + 1.93i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.607 + 0.441i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.433 - 1.33i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.42 - 4.38i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.000945 - 0.00291i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-4.80 + 3.49i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-2.92 + 2.12i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.30 + 4.02i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-0.140 + 0.101i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.52 - 6.92i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.51 + 4.67i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.221 - 0.680i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 3.00T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 5.64T + 83T^{2} \)
89 \( 1 + (-6.96 - 5.05i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.25 + 3.87i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.31352048128746540828267134862, −8.902177475413841660613522276165, −8.149739855219225304443052721386, −7.10802499341158054418885941516, −6.36082633229756115381192875790, −5.71629957731971964757891084970, −4.88856528941858050455802175823, −3.57674397141885220554801927450, −2.64132473334697278697551735204, −0.68104146353334924239310197644, 0.76024271152591266876462171594, 2.08667870609198114694739288145, 4.12903659926117063579162202095, 4.65729417312033135397257905054, 5.40115629380907219591834149629, 6.28797002775381116631747841578, 7.29286381688346340456840191746, 8.041221171479625366781383054955, 9.001874277124579925696576659210, 9.920477629897124935104825118543

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