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Properties

Label 2-1148-41.16-c1-0-5
Degree $2$
Conductor $1148$
Sign $0.626 - 0.779i$
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  − 3.15·3-s + (1.01 + 0.736i)5-s + (0.309 + 0.951i)7-s + 6.95·9-s + (3.45 − 2.50i)11-s + (−1.21 + 3.75i)13-s + (−3.20 − 2.32i)15-s + (−4.64 + 3.37i)17-s + (−1.20 − 3.69i)19-s + (−0.975 − 3.00i)21-s + (1.39 − 4.28i)23-s + (−1.05 − 3.26i)25-s − 12.4·27-s + (5.73 + 4.16i)29-s + (7.70 − 5.60i)31-s + ⋯
L(s)  = 1  − 1.82·3-s + (0.453 + 0.329i)5-s + (0.116 + 0.359i)7-s + 2.31·9-s + (1.04 − 0.756i)11-s + (−0.337 + 1.04i)13-s + (−0.826 − 0.600i)15-s + (−1.12 + 0.817i)17-s + (−0.275 − 0.848i)19-s + (−0.212 − 0.654i)21-s + (0.290 − 0.892i)23-s + (−0.211 − 0.652i)25-s − 2.40·27-s + (1.06 + 0.774i)29-s + (1.38 − 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9133154686\)
\(L(\frac12)\) \(\approx\) \(0.9133154686\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.01 - 4.98i)T \)
good3 \( 1 + 3.15T + 3T^{2} \)
5 \( 1 + (-1.01 - 0.736i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (-3.45 + 2.50i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.21 - 3.75i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.64 - 3.37i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.20 + 3.69i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.39 + 4.28i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-5.73 - 4.16i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.70 + 5.60i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.60 + 2.61i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (3.40 - 10.4i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (0.993 - 3.05i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.84 - 1.34i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.664 - 2.04i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.86 - 11.9i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-7.13 - 5.18i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (10.5 - 7.69i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 - 4.27T + 83T^{2} \)
89 \( 1 + (0.204 + 0.630i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.16 - 3.75i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.13171120434491686760480656964, −9.249740193800039410324529296551, −8.385193972418253052129100944078, −6.80024903324410449904812260908, −6.49274173199845729099979570352, −5.98102503233615915517873499326, −4.70995991555551106562536874693, −4.27098516024591668324621876651, −2.39841847088940807407743705333, −1.01414980951396649101519438452, 0.66231838659730085719090869081, 1.83399285584661394951962602155, 3.79986927755557972625820357926, 4.89780476342087869389121381214, 5.24505641449155922804234173669, 6.41961267182818384692790919267, 6.81129612299898611618434792270, 7.81713699911926091902572242998, 9.129756448581705602130027384618, 9.982273514866493488093971051709

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