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Properties

Label 2-1148-41.10-c1-0-4
Degree $2$
Conductor $1148$
Sign $-0.469 - 0.883i$
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  + 1.51·3-s + (0.929 + 2.86i)5-s + (−0.809 + 0.587i)7-s − 0.703·9-s + (−0.853 + 2.62i)11-s + (−5.09 − 3.70i)13-s + (1.40 + 4.33i)15-s + (−1.66 + 5.12i)17-s + (0.116 − 0.0843i)19-s + (−1.22 + 0.890i)21-s + (5.98 + 4.34i)23-s + (−3.27 + 2.38i)25-s − 5.61·27-s + (2.17 + 6.68i)29-s + (−0.0687 + 0.211i)31-s + ⋯
L(s)  = 1  + 0.874·3-s + (0.415 + 1.27i)5-s + (−0.305 + 0.222i)7-s − 0.234·9-s + (−0.257 + 0.791i)11-s + (−1.41 − 1.02i)13-s + (0.363 + 1.11i)15-s + (−0.403 + 1.24i)17-s + (0.0266 − 0.0193i)19-s + (−0.267 + 0.194i)21-s + (1.24 + 0.906i)23-s + (−0.655 + 0.476i)25-s − 1.08·27-s + (0.403 + 1.24i)29-s + (−0.0123 + 0.0379i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.469 - 0.883i$
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: \((2,\ 1148,\ (\ :1/2),\ -0.469 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.636202788\)
\(L(\frac12)\) \(\approx\) \(1.636202788\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.33 + 0.910i)T \)
good3 \( 1 - 1.51T + 3T^{2} \)
5 \( 1 + (-0.929 - 2.86i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (0.853 - 2.62i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (5.09 + 3.70i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.66 - 5.12i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.116 + 0.0843i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.98 - 4.34i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.17 - 6.68i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.0687 - 0.211i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.744 + 2.29i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (3.24 + 2.35i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-0.527 - 0.383i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.68 + 5.18i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.54 - 1.12i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.168 - 0.122i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.547 + 1.68i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (4.31 - 13.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 2.49T + 79T^{2} \)
83 \( 1 + 4.79T + 83T^{2} \)
89 \( 1 + (-0.0253 + 0.0184i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.12 - 9.61i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.08025235072855467482536114306, −9.322378589492061693100818501240, −8.453728798411277674135273041977, −7.45820344422014643053699263387, −7.00536753481340635249743026520, −5.90228323499179231511301471335, −5.00364873897142103458537470785, −3.50910138377755816249309213784, −2.82117305935953899253685022891, −2.10032066293881274629050917273, 0.59099004040823624631988463731, 2.24000711667238698325906690064, 3.02263496551838287213336709511, 4.47773013165697849973115847283, 5.00749268759965447380999190855, 6.14723469825416743244433716484, 7.19768183486981271133486444280, 8.062988848807482202356026033633, 8.974335333924632639148600642387, 9.222013943535689696086935502645

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