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Properties

Label 2-1148-287.81-c1-0-0
Degree $2$
Conductor $1148$
Sign $-0.401 - 0.915i$
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  + (0.294 + 0.169i)3-s + (−2.09 − 3.62i)5-s + (−0.0873 + 2.64i)7-s + (−1.44 − 2.49i)9-s + (−3.02 − 1.74i)11-s + 4.69i·13-s − 1.42i·15-s + (0.513 + 0.296i)17-s + (0.364 − 0.210i)19-s + (−0.474 + 0.762i)21-s + (2.40 + 4.17i)23-s + (−6.25 + 10.8i)25-s − 1.99i·27-s + 5.24i·29-s + (−0.176 + 0.306i)31-s + ⋯
L(s)  = 1  + (0.169 + 0.0980i)3-s + (−0.935 − 1.62i)5-s + (−0.0329 + 0.999i)7-s + (−0.480 − 0.832i)9-s + (−0.912 − 0.526i)11-s + 1.30i·13-s − 0.366i·15-s + (0.124 + 0.0719i)17-s + (0.0835 − 0.0482i)19-s + (−0.103 + 0.166i)21-s + (0.502 + 0.870i)23-s + (−1.25 + 2.16i)25-s − 0.384i·27-s + 0.974i·29-s + (−0.0317 + 0.0550i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.401 - 0.915i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.401 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3970976565\)
\(L(\frac12)\) \(\approx\) \(0.3970976565\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0873 - 2.64i)T \)
41 \( 1 + (3.27 - 5.49i)T \)
good3 \( 1 + (-0.294 - 0.169i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.09 + 3.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.02 + 1.74i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.69iT - 13T^{2} \)
17 \( 1 + (-0.513 - 0.296i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.364 + 0.210i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.40 - 4.17i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.24iT - 29T^{2} \)
31 \( 1 + (0.176 - 0.306i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.33 - 2.31i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 1.13T + 43T^{2} \)
47 \( 1 + (-2.07 + 1.20i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.65 + 3.26i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.0567 + 0.0982i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.54 - 2.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.74 - 3.31i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.06iT - 71T^{2} \)
73 \( 1 + (2.25 - 3.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (11.1 - 6.41i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.34T + 83T^{2} \)
89 \( 1 + (5.69 - 3.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−9.638202142797421348456121544009, −9.093943376593857839506024869259, −8.531581453823145081104149934970, −7.944229860648382734987605544860, −6.72716957854468730792305049936, −5.56059191059544883287854129249, −5.00735097551833150693088675202, −3.96424639255001615179381872997, −2.98882944370375963383202212456, −1.39629367818367310182852985843, 0.17176552249601207076163068249, 2.47492505138184122357764583304, 3.11367129584896872628200408676, 4.12528702422228929829133484930, 5.22028172236558238561490652439, 6.39267125644333549190546484405, 7.35983864643990277639705171673, 7.69376696453059682766880957971, 8.318493916600150557071997707101, 9.930004886968004781340975355759

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