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Properties

Label 2-1148-41.32-c1-0-16
Degree $2$
Conductor $1148$
Sign $0.767 + 0.640i$
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  + (1.86 − 1.86i)3-s + 2.96i·5-s + (0.707 − 0.707i)7-s − 3.95i·9-s + (2.99 − 2.99i)11-s + (0.619 − 0.619i)13-s + (5.52 + 5.52i)15-s + (1.08 + 1.08i)17-s + (−1.93 − 1.93i)19-s − 2.63i·21-s − 1.66·23-s − 3.77·25-s + (−1.77 − 1.77i)27-s + (−2.85 + 2.85i)29-s + 10.4·31-s + ⋯
L(s)  = 1  + (1.07 − 1.07i)3-s + 1.32i·5-s + (0.267 − 0.267i)7-s − 1.31i·9-s + (0.901 − 0.901i)11-s + (0.171 − 0.171i)13-s + (1.42 + 1.42i)15-s + (0.262 + 0.262i)17-s + (−0.444 − 0.444i)19-s − 0.575i·21-s − 0.347·23-s − 0.755·25-s + (−0.341 − 0.341i)27-s + (−0.529 + 0.529i)29-s + 1.87·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.767 + 0.640i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.767 + 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.569023598\)
\(L(\frac12)\) \(\approx\) \(2.569023598\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (2.65 + 5.82i)T \)
good3 \( 1 + (-1.86 + 1.86i)T - 3iT^{2} \)
5 \( 1 - 2.96iT - 5T^{2} \)
11 \( 1 + (-2.99 + 2.99i)T - 11iT^{2} \)
13 \( 1 + (-0.619 + 0.619i)T - 13iT^{2} \)
17 \( 1 + (-1.08 - 1.08i)T + 17iT^{2} \)
19 \( 1 + (1.93 + 1.93i)T + 19iT^{2} \)
23 \( 1 + 1.66T + 23T^{2} \)
29 \( 1 + (2.85 - 2.85i)T - 29iT^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 - 5.67T + 37T^{2} \)
43 \( 1 + 0.246iT - 43T^{2} \)
47 \( 1 + (-1.86 - 1.86i)T + 47iT^{2} \)
53 \( 1 + (2.37 - 2.37i)T - 53iT^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 2.07iT - 61T^{2} \)
67 \( 1 + (3.35 + 3.35i)T + 67iT^{2} \)
71 \( 1 + (7.87 - 7.87i)T - 71iT^{2} \)
73 \( 1 + 2.12iT - 73T^{2} \)
79 \( 1 + (3.02 - 3.02i)T - 79iT^{2} \)
83 \( 1 + 2.75T + 83T^{2} \)
89 \( 1 + (9.71 - 9.71i)T - 89iT^{2} \)
97 \( 1 + (11.1 + 11.1i)T + 97iT^{2} \)
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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

−9.617174694634528695547911552435, −8.584453731605698407277698188098, −8.119877963032433367544980269714, −7.13968817281703458980864512973, −6.68304573889992761348893076696, −5.85395601314054905980005239173, −4.10274623695299957633943547965, −3.18605250028760892128184395299, −2.47135089751663908621284527794, −1.20827878559948861420496787859, 1.44149443300788609744125895549, 2.67705193469657134699580904532, 4.09812362353426192943038919957, 4.36966256322993305505535194815, 5.31643530248204336102597541571, 6.51333320020057554782404221965, 7.902983939260205620899223589546, 8.410065081007636602748994767305, 9.148370189697195129045715817045, 9.663284022709207966046937764189

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