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Properties

Label 2-1148-41.10-c1-0-3
Degree $2$
Conductor $1148$
Sign $-0.933 - 0.358i$
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  − 2.40·3-s + (1.08 + 3.34i)5-s + (−0.809 + 0.587i)7-s + 2.80·9-s + (−1.11 + 3.42i)11-s + (4.63 + 3.36i)13-s + (−2.62 − 8.06i)15-s + (0.00444 − 0.0136i)17-s + (0.791 − 0.575i)19-s + (1.94 − 1.41i)21-s + (2.12 + 1.54i)23-s + (−5.98 + 4.34i)25-s + 0.478·27-s + (−0.429 − 1.32i)29-s + (−1.33 + 4.10i)31-s + ⋯
L(s)  = 1  − 1.39·3-s + (0.486 + 1.49i)5-s + (−0.305 + 0.222i)7-s + 0.933·9-s + (−0.335 + 1.03i)11-s + (1.28 + 0.933i)13-s + (−0.676 − 2.08i)15-s + (0.00107 − 0.00331i)17-s + (0.181 − 0.132i)19-s + (0.425 − 0.308i)21-s + (0.443 + 0.322i)23-s + (−1.19 + 0.869i)25-s + 0.0919·27-s + (−0.0796 − 0.245i)29-s + (−0.239 + 0.737i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.933 - 0.358i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.933 - 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8053668436\)
\(L(\frac12)\) \(\approx\) \(0.8053668436\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.18 + 4.84i)T \)
good3 \( 1 + 2.40T + 3T^{2} \)
5 \( 1 + (-1.08 - 3.34i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (1.11 - 3.42i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-4.63 - 3.36i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.00444 + 0.0136i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.791 + 0.575i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.12 - 1.54i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.429 + 1.32i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.33 - 4.10i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.05 - 3.23i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (9.78 + 7.10i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (6.98 + 5.07i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.40 - 10.4i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.98 - 3.62i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.0940 + 0.0683i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.918 + 2.82i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.84 - 11.8i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 3.56T + 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 - 6.79T + 83T^{2} \)
89 \( 1 + (11.2 - 8.17i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.29 + 13.2i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.33825271781935464235800349510, −9.694376244233131278274516568134, −8.610688543206431664014096216943, −7.04688432027526542303095103443, −6.89231888671094480951162004090, −6.01549415598187904974660904550, −5.33761018031386244459570487738, −4.13827160881274651266243414938, −2.95082214190549789544210464811, −1.68323484936585510209649468224, 0.47052142577224131059597641197, 1.28892832228080427642032073975, 3.27011562090309384549433594396, 4.52575340314199834819478103249, 5.37796119376013885994706945753, 5.84818682551289623830626220696, 6.52979402711776436712618831912, 8.004538147069605055520923325496, 8.572136955465328552315140105683, 9.516819328708257905366035088220

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