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Properties

Label 2-1148-41.31-c1-0-6
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  + 0.304i·3-s + (−0.420 − 1.29i)5-s + (0.587 + 0.809i)7-s + 2.90·9-s + (−1.66 − 0.539i)11-s + (−3.66 + 5.03i)13-s + (0.395 − 0.128i)15-s + (1.26 + 0.412i)17-s + (1.99 + 2.74i)19-s + (−0.246 + 0.179i)21-s + (2.91 + 2.11i)23-s + (2.54 − 1.84i)25-s + 1.80i·27-s + (5.01 − 1.62i)29-s + (−0.893 + 2.74i)31-s + ⋯
L(s)  = 1  + 0.176i·3-s + (−0.188 − 0.579i)5-s + (0.222 + 0.305i)7-s + 0.969·9-s + (−0.500 − 0.162i)11-s + (−1.01 + 1.39i)13-s + (0.101 − 0.0331i)15-s + (0.307 + 0.100i)17-s + (0.457 + 0.629i)19-s + (−0.0538 + 0.0391i)21-s + (0.607 + 0.441i)23-s + (0.508 − 0.369i)25-s + 0.346i·27-s + (0.930 − 0.302i)29-s + (−0.160 + 0.493i)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: $\chi_{1148} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.767 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.608164432\)
\(L(\frac12)\) \(\approx\) \(1.608164432\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-3.76 - 5.17i)T \)
good3 \( 1 - 0.304iT - 3T^{2} \)
5 \( 1 + (0.420 + 1.29i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (1.66 + 0.539i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.66 - 5.03i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.26 - 0.412i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.99 - 2.74i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.91 - 2.11i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-5.01 + 1.62i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.893 - 2.74i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.584 + 1.79i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (2.46 + 1.79i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-2.78 + 3.83i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.78 + 1.22i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-11.2 - 8.16i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.00534 - 0.00388i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (7.38 - 2.40i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.97 + 0.641i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 - 7.16T + 73T^{2} \)
79 \( 1 - 14.2iT - 79T^{2} \)
83 \( 1 + 6.99T + 83T^{2} \)
89 \( 1 + (-8.41 - 11.5i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-10.1 + 3.30i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.845552694011988349322897354773, −9.129298280014821913737948460488, −8.311663653159689494604800749254, −7.39537459029244520924516833657, −6.71302478170302661635203636790, −5.41248993236354609576440303199, −4.73051371170358780923491297702, −3.93060141594993026574784024444, −2.51698684533525904112877508227, −1.26826905920887531025584911975, 0.818442325103169222236262208538, 2.46568680217749147086439974630, 3.32637161352859101198874430740, 4.64513700927798653406136226991, 5.28703365088514240640426899781, 6.57984887249625376673433014892, 7.43274158833316980533504897861, 7.68556081539539247417333805882, 8.892461461100380924440421082870, 10.00676135034113451895134113623

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