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Properties

Label 2-1148-41.16-c1-0-2
Degree $2$
Conductor $1148$
Sign $0.867 - 0.497i$
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  − 1.54·3-s + (−2.70 − 1.96i)5-s + (−0.309 − 0.951i)7-s − 0.607·9-s + (−3.53 + 2.56i)11-s + (0.0240 − 0.0739i)13-s + (4.18 + 3.04i)15-s + (−3.97 + 2.89i)17-s + (−0.396 − 1.21i)19-s + (0.477 + 1.47i)21-s + (1.68 − 5.19i)23-s + (1.91 + 5.89i)25-s + 5.57·27-s + (2.49 + 1.81i)29-s + (2.46 − 1.79i)31-s + ⋯
L(s)  = 1  − 0.893·3-s + (−1.21 − 0.879i)5-s + (−0.116 − 0.359i)7-s − 0.202·9-s + (−1.06 + 0.774i)11-s + (0.00666 − 0.0205i)13-s + (1.08 + 0.785i)15-s + (−0.965 + 0.701i)17-s + (−0.0909 − 0.279i)19-s + (0.104 + 0.321i)21-s + (0.351 − 1.08i)23-s + (0.383 + 1.17i)25-s + 1.07·27-s + (0.463 + 0.336i)29-s + (0.442 − 0.321i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.867 - 0.497i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.867 - 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4393138349\)
\(L(\frac12)\) \(\approx\) \(0.4393138349\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.92 + 6.10i)T \)
good3 \( 1 + 1.54T + 3T^{2} \)
5 \( 1 + (2.70 + 1.96i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (3.53 - 2.56i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.0240 + 0.0739i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.97 - 2.89i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.396 + 1.21i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.68 + 5.19i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.49 - 1.81i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.46 + 1.79i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.87 - 1.36i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (2.58 - 7.96i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-4.12 + 12.6i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.76 - 5.64i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.864 + 2.66i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.90 - 12.0i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-7.78 - 5.65i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.89 - 3.55i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 - 2.82T + 73T^{2} \)
79 \( 1 + 4.09T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 + (-1.67 - 5.16i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.71 - 4.14i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.16862815264475019651336216926, −8.729000075719672540583609936132, −8.378270474883716200958610733781, −7.34201496066841613063540474367, −6.59849675921550593075893856085, −5.43466376001760915035166367055, −4.68458301475069744261936677731, −4.06908381306486566263438131427, −2.55947328115166002656704353310, −0.70144742584042937497210700969, 0.35415492955342878814600490169, 2.64308599781574986623632157722, 3.39968613395487615121572852732, 4.65238190627732114781262364088, 5.52883263989055028448301740150, 6.38208768515003093004110295298, 7.19028547691937613812730508780, 8.024436860337444880003024278839, 8.747928082144615551270668727758, 9.965097277525290797618132222526

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