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Properties

Label 2-1148-41.10-c1-0-8
Degree $2$
Conductor $1148$
Sign $0.286 - 0.957i$
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  + 2.41·3-s + (0.746 + 2.29i)5-s + (−0.809 + 0.587i)7-s + 2.82·9-s + (−1.5 + 4.61i)11-s + (4.36 + 3.17i)13-s + (1.80 + 5.54i)15-s + (1.33 − 4.10i)17-s + (−3.41 + 2.48i)19-s + (−1.95 + 1.41i)21-s + (−7.43 − 5.40i)23-s + (−0.670 + 0.486i)25-s − 0.414·27-s + (−0.268 − 0.827i)29-s + (−0.286 + 0.883i)31-s + ⋯
L(s)  = 1  + 1.39·3-s + (0.333 + 1.02i)5-s + (−0.305 + 0.222i)7-s + 0.942·9-s + (−0.452 + 1.39i)11-s + (1.21 + 0.880i)13-s + (0.465 + 1.43i)15-s + (0.323 − 0.995i)17-s + (−0.784 + 0.569i)19-s + (−0.426 + 0.309i)21-s + (−1.55 − 1.12i)23-s + (−0.134 + 0.0973i)25-s − 0.0797·27-s + (−0.0499 − 0.153i)29-s + (−0.0515 + 0.158i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.286 - 0.957i$
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: \((2,\ 1148,\ (\ :1/2),\ 0.286 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.583986440\)
\(L(\frac12)\) \(\approx\) \(2.583986440\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.14 - 3.80i)T \)
good3 \( 1 - 2.41T + 3T^{2} \)
5 \( 1 + (-0.746 - 2.29i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (1.5 - 4.61i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-4.36 - 3.17i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.33 + 4.10i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.41 - 2.48i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (7.43 + 5.40i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.268 + 0.827i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.286 - 0.883i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.260 + 0.802i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-4.79 - 3.48i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-8.67 - 6.30i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.26 + 13.1i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.90 - 3.56i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-12.1 + 8.81i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.68 - 8.25i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.77 + 8.54i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 3.15T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (-10.6 + 7.72i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.50 - 4.62i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.871189752109862854930687059913, −9.200274764338604154191612152557, −8.331337500618729115651082784919, −7.59566886774284804537200576912, −6.72698650361474009434079691550, −6.03115860782378443644473358279, −4.46474231012381240764359543639, −3.64871356446489254198472412702, −2.52627745098380081117728925315, −2.05770059831008913875363552081, 0.973116768581227309438874224101, 2.29600867456064851261085254149, 3.49000151484456243843801596890, 3.98329604623302409535186936970, 5.59679416254560099656004729110, 5.99730075486022347127668312967, 7.55255328133638891058714764676, 8.310161115013631514537609245418, 8.658751029462908535662536304861, 9.324826255364799465675991733802

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