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Properties

Label 2-1148-41.37-c1-0-13
Degree $2$
Conductor $1148$
Sign $0.987 - 0.158i$
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  + 2.72·3-s + (−0.323 + 0.994i)5-s + (0.809 + 0.587i)7-s + 4.43·9-s + (−1.27 − 3.93i)11-s + (1.98 − 1.43i)13-s + (−0.881 + 2.71i)15-s + (1.62 + 5.01i)17-s + (0.543 + 0.394i)19-s + (2.20 + 1.60i)21-s + (5.08 − 3.69i)23-s + (3.15 + 2.29i)25-s + 3.91·27-s + (0.554 − 1.70i)29-s + (−0.299 − 0.920i)31-s + ⋯
L(s)  = 1  + 1.57·3-s + (−0.144 + 0.444i)5-s + (0.305 + 0.222i)7-s + 1.47·9-s + (−0.385 − 1.18i)11-s + (0.549 − 0.399i)13-s + (−0.227 + 0.700i)15-s + (0.395 + 1.21i)17-s + (0.124 + 0.0906i)19-s + (0.481 + 0.349i)21-s + (1.06 − 0.770i)23-s + (0.631 + 0.459i)25-s + 0.752·27-s + (0.103 − 0.317i)29-s + (−0.0537 − 0.165i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.987 - 0.158i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.987 - 0.158i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.917484224\)
\(L(\frac12)\) \(\approx\) \(2.917484224\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.09 - 5.60i)T \)
good3 \( 1 - 2.72T + 3T^{2} \)
5 \( 1 + (0.323 - 0.994i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (1.27 + 3.93i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.98 + 1.43i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.62 - 5.01i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.543 - 0.394i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-5.08 + 3.69i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.554 + 1.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.299 + 0.920i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.475 - 1.46i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-1.58 + 1.15i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (7.27 - 5.28i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.322 + 0.992i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.41 - 6.84i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.91 + 5.74i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.76 - 8.51i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.81 + 11.7i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 - 5.61T + 79T^{2} \)
83 \( 1 + 15.9T + 83T^{2} \)
89 \( 1 + (0.0547 + 0.0397i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.922 - 2.83i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.614759785352186957609852794620, −8.798406850944195276183846475782, −8.216378138916315221384243327378, −7.75159835149424639224340853641, −6.58247297512511515756264834995, −5.65106738714997021836255469128, −4.33600896328920391276622395984, −3.22078464516334555860705687980, −2.88779829795173420191896456129, −1.42680581820110607348400687640, 1.38368522325938993894395274870, 2.51515156263484145858166682288, 3.44545761353787252498888776951, 4.48540122905962194759851682742, 5.20398585020055148664641454089, 6.87286496084790914082699593819, 7.46644891697953355077621338962, 8.160751514009235641364196250568, 9.084319213439044263617271172707, 9.408064297801522251202053747577

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