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Properties

Label 2-1148-41.18-c1-0-9
Degree $2$
Conductor $1148$
Sign $0.495 - 0.868i$
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  + 2.41·3-s + (−1.95 + 1.41i)5-s + (0.309 − 0.951i)7-s + 2.82·9-s + (−1.5 − 1.08i)11-s + (1.66 + 5.13i)13-s + (−4.71 + 3.42i)15-s + (4.49 + 3.26i)17-s + (−2.03 + 6.24i)19-s + (0.746 − 2.29i)21-s + (0.195 + 0.602i)23-s + (0.255 − 0.787i)25-s − 0.414·27-s + (4.01 − 2.91i)29-s + (8.73 + 6.34i)31-s + ⋯
L(s)  = 1  + 1.39·3-s + (−0.873 + 0.634i)5-s + (0.116 − 0.359i)7-s + 0.942·9-s + (−0.452 − 0.328i)11-s + (0.462 + 1.42i)13-s + (−1.21 + 0.884i)15-s + (1.09 + 0.792i)17-s + (−0.465 + 1.43i)19-s + (0.162 − 0.501i)21-s + (0.0408 + 0.125i)23-s + (0.0511 − 0.157i)25-s − 0.0797·27-s + (0.744 − 0.541i)29-s + (1.56 + 1.14i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.495 - 0.868i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.495 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.192640922\)
\(L(\frac12)\) \(\approx\) \(2.192640922\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.11 + 1.90i)T \)
good3 \( 1 - 2.41T + 3T^{2} \)
5 \( 1 + (1.95 - 1.41i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (1.5 + 1.08i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.66 - 5.13i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.49 - 3.26i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.03 - 6.24i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.195 - 0.602i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.01 + 2.91i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-8.73 - 6.34i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-9.41 + 6.84i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (2.64 + 8.13i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.265 + 0.815i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.43 + 1.77i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.74 - 5.37i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.82 - 5.60i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (4.02 - 2.92i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-1.46 - 1.06i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 4.10T + 73T^{2} \)
79 \( 1 + 7.64T + 79T^{2} \)
83 \( 1 - 1.94T + 83T^{2} \)
89 \( 1 + (-3.46 + 10.6i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (6.05 - 4.39i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.02456745872870949098368792432, −8.840403485682319530560729947134, −8.261502376314387153076417731640, −7.71138057927704000702837950602, −6.87786404383904371639574162452, −5.84959800832749557375663015772, −4.21732584010481705173701563125, −3.73474090753273285649635093378, −2.86581196896759497786744876080, −1.62003481025621656182120372094, 0.863081037926993436670573078121, 2.68791356622081199669613683659, 3.10836948479747805320426257782, 4.40937008166524498846979100972, 5.10030801674584125471936423887, 6.42935676740106565317414941626, 7.80428940365747234106468150305, 7.972976525868272598477512736889, 8.620644168536240147027733233800, 9.529271218598962029770040075725

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