Properties

Label 2-1148-287.163-c1-0-8
Degree $2$
Conductor $1148$
Sign $0.314 - 0.949i$
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  + (−1.32 + 0.762i)3-s + (0.729 − 1.26i)5-s + (0.676 − 2.55i)7-s + (−0.337 + 0.584i)9-s + (−3.72 + 2.14i)11-s − 0.589i·13-s + 2.22i·15-s + (−6.82 + 3.94i)17-s + (6.95 + 4.01i)19-s + (1.05 + 3.89i)21-s + (3.03 − 5.24i)23-s + (1.43 + 2.48i)25-s − 5.60i·27-s + 1.42i·29-s + (4.04 + 7.00i)31-s + ⋯
L(s)  = 1  + (−0.762 + 0.440i)3-s + (0.326 − 0.565i)5-s + (0.255 − 0.966i)7-s + (−0.112 + 0.194i)9-s + (−1.12 + 0.648i)11-s − 0.163i·13-s + 0.574i·15-s + (−1.65 + 0.956i)17-s + (1.59 + 0.920i)19-s + (0.230 + 0.849i)21-s + (0.631 − 1.09i)23-s + (0.287 + 0.497i)25-s − 1.07i·27-s + 0.264i·29-s + (0.726 + 1.25i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.314 - 0.949i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9939206720\)
\(L(\frac12)\) \(\approx\) \(0.9939206720\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.676 + 2.55i)T \)
41 \( 1 + (-5.10 - 3.86i)T \)
good3 \( 1 + (1.32 - 0.762i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.729 + 1.26i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.72 - 2.14i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.589iT - 13T^{2} \)
17 \( 1 + (6.82 - 3.94i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.95 - 4.01i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.03 + 5.24i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.42iT - 29T^{2} \)
31 \( 1 + (-4.04 - 7.00i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.05 - 1.82i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 11.6T + 43T^{2} \)
47 \( 1 + (-1.69 - 0.978i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.19 - 0.691i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.36 + 2.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.43 - 9.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.45 - 0.838i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.55iT - 71T^{2} \)
73 \( 1 + (-1.00 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.8 - 7.40i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.67T + 83T^{2} \)
89 \( 1 + (-0.839 - 0.484i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.46iT - 97T^{2} \)
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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

−10.29177033819005045662608042774, −9.252481489544090462799147954088, −8.280468473953204916444887275737, −7.52637502254477345583308743490, −6.55767270106860031286811414577, −5.48872752135750808554918106632, −4.86239128175776496479207211394, −4.19432121780891102082197131923, −2.65692350350400677445939859047, −1.18422190645236671742185211173, 0.53482850288620895811398061614, 2.37184384883452290140211261166, 3.02147377085380653623845678223, 4.76708700079095718566567962060, 5.54092973112083593664268127501, 6.16072641309026581236795327145, 7.06532555791545540963455626951, 7.81111137495592592947986216423, 9.113352871628054782826179876715, 9.363573208569216943045464975568

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