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Properties

Label 2-1148-41.31-c1-0-8
Degree $2$
Conductor $1148$
Sign $0.958 + 0.284i$
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  − 0.904i·3-s + (0.129 + 0.398i)5-s + (−0.587 − 0.809i)7-s + 2.18·9-s + (1.99 + 0.646i)11-s + (−1.84 + 2.54i)13-s + (0.360 − 0.117i)15-s + (0.218 + 0.0711i)17-s + (1.62 + 2.24i)19-s + (−0.731 + 0.531i)21-s + (6.51 + 4.73i)23-s + (3.90 − 2.83i)25-s − 4.68i·27-s + (−6.06 + 1.97i)29-s + (1.52 − 4.70i)31-s + ⋯
L(s)  = 1  − 0.522i·3-s + (0.0579 + 0.178i)5-s + (−0.222 − 0.305i)7-s + 0.727·9-s + (0.600 + 0.195i)11-s + (−0.511 + 0.704i)13-s + (0.0930 − 0.0302i)15-s + (0.0531 + 0.0172i)17-s + (0.373 + 0.514i)19-s + (−0.159 + 0.115i)21-s + (1.35 + 0.986i)23-s + (0.780 − 0.567i)25-s − 0.901i·27-s + (−1.12 + 0.365i)29-s + (0.274 − 0.845i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.958 + 0.284i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.958 + 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.789454922\)
\(L(\frac12)\) \(\approx\) \(1.789454922\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 6.00i)T \)
good3 \( 1 + 0.904iT - 3T^{2} \)
5 \( 1 + (-0.129 - 0.398i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-1.99 - 0.646i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.84 - 2.54i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.218 - 0.0711i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.62 - 2.24i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-6.51 - 4.73i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (6.06 - 1.97i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.52 + 4.70i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.51 + 4.66i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-8.75 - 6.36i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-0.514 + 0.707i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.28 - 0.417i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.10 - 0.799i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.01 - 1.46i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.04 + 0.987i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-9.37 - 3.04i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 + 9.73iT - 79T^{2} \)
83 \( 1 - 4.97T + 83T^{2} \)
89 \( 1 + (3.59 + 4.95i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (13.7 - 4.48i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.516388647590747606148475235571, −9.254217681814529449979155611295, −7.86720580919365133917247176229, −7.18429970992464721957870992260, −6.68933817697575342424239232886, −5.60595343063562963224185419399, −4.46942718739320634421446494919, −3.61363688052076114667763510890, −2.24772335301655586410655119642, −1.10764614403616547962291773528, 1.06287821916242852756345317797, 2.68102158558954545957617510857, 3.65842188269420687266995578564, 4.76429376121874603047112718767, 5.35860154666399604335059230183, 6.60894033859953270855646653272, 7.24270891056407415112110599897, 8.346889589681250717565214987112, 9.245533146670306126097394029986, 9.646910128222463168392144951429

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