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Properties

Label 2-1148-41.10-c1-0-10
Degree $2$
Conductor $1148$
Sign $0.159 + 0.987i$
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 + (−0.127 − 0.393i)5-s + (−0.809 + 0.587i)7-s − 2.82·9-s + (−1.5 + 4.61i)11-s + (−0.749 − 0.544i)13-s + (0.0530 + 0.163i)15-s + (1.66 − 5.13i)17-s + (6.27 − 4.55i)19-s + (0.335 − 0.243i)21-s + (−0.0336 − 0.0244i)23-s + (3.90 − 2.83i)25-s + 2.41·27-s + (−1.34 − 4.15i)29-s + (2.66 − 8.21i)31-s + ⋯
L(s)  = 1  − 0.239·3-s + (−0.0572 − 0.176i)5-s + (−0.305 + 0.222i)7-s − 0.942·9-s + (−0.452 + 1.39i)11-s + (−0.207 − 0.150i)13-s + (0.0136 + 0.0421i)15-s + (0.404 − 1.24i)17-s + (1.43 − 1.04i)19-s + (0.0731 − 0.0531i)21-s + (−0.00700 − 0.00509i)23-s + (0.781 − 0.567i)25-s + 0.464·27-s + (−0.250 − 0.771i)29-s + (0.479 − 1.47i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.159 + 0.987i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (953, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9551243597\)
\(L(\frac12)\) \(\approx\) \(0.9551243597\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.29 + 1.17i)T \)
good3 \( 1 + 0.414T + 3T^{2} \)
5 \( 1 + (0.127 + 0.393i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (1.5 - 4.61i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.749 + 0.544i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.66 + 5.13i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.27 + 4.55i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.0336 + 0.0244i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.34 + 4.15i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.66 + 8.21i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.406 - 1.25i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (7.18 + 5.21i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-7.26 - 5.27i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.43 + 4.43i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.90 + 2.84i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (8.66 - 6.29i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.97 + 9.15i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.69 + 5.21i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 3.73T + 73T^{2} \)
79 \( 1 - 5.31T + 79T^{2} \)
83 \( 1 - 8.05T + 83T^{2} \)
89 \( 1 + (1.54 - 1.12i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.29 - 3.98i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.597992151962479550445969973682, −8.979677844965946441359155647656, −7.81585573818682287646506276748, −7.24085486412693818990219577837, −6.22631312157025351836508637099, −5.17295636445456495507181286072, −4.72341776727810070286390529749, −3.15128946142851904080219341249, −2.36541865546306723258829293628, −0.46549052152109177377426030586, 1.25838111239458487831152730549, 3.10906898396424971044030434338, 3.45632139606963730954208177594, 5.09332098324997091111126737570, 5.74333369640518420996923691280, 6.50816370365971879615925128897, 7.59310424340512944164410676920, 8.393206208377380204323713790112, 9.020552087642027849832642578448, 10.23898925254279266792936841956

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