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Properties

Label 2-737-737.406-c0-0-0
Degree $2$
Conductor $737$
Sign $0.982 + 0.187i$
Analytic cond. $0.367810$
Root an. cond. $0.606474$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.698 + 0.449i)3-s + (0.981 + 0.189i)4-s + (−0.205 − 1.43i)5-s + (−0.128 − 0.281i)9-s + (−0.786 + 0.618i)11-s + (0.601 + 0.573i)12-s + (0.499 − 1.09i)15-s + (0.928 + 0.371i)16-s + (0.0688 − 1.44i)20-s + (−1.49 + 0.770i)23-s + (−1.05 + 0.308i)25-s + (0.154 − 1.07i)27-s + (0.273 + 1.12i)31-s + (−0.827 + 0.0789i)33-s + (−0.0730 − 0.300i)36-s + (−0.580 + 1.00i)37-s + ⋯
L(s)  = 1  + (0.698 + 0.449i)3-s + (0.981 + 0.189i)4-s + (−0.205 − 1.43i)5-s + (−0.128 − 0.281i)9-s + (−0.786 + 0.618i)11-s + (0.601 + 0.573i)12-s + (0.499 − 1.09i)15-s + (0.928 + 0.371i)16-s + (0.0688 − 1.44i)20-s + (−1.49 + 0.770i)23-s + (−1.05 + 0.308i)25-s + (0.154 − 1.07i)27-s + (0.273 + 1.12i)31-s + (−0.827 + 0.0789i)33-s + (−0.0730 − 0.300i)36-s + (−0.580 + 1.00i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 737 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(737\)    =    \(11 \cdot 67\)
Sign: $0.982 + 0.187i$
Analytic conductor: \(0.367810\)
Root analytic conductor: \(0.606474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{737} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 737,\ (\ :0),\ 0.982 + 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.322355883\)
\(L(\frac12)\) \(\approx\) \(1.322355883\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.786 - 0.618i)T \)
67 \( 1 + (0.888 + 0.458i)T \)
good2 \( 1 + (-0.981 - 0.189i)T^{2} \)
3 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
5 \( 1 + (0.205 + 1.43i)T + (-0.959 + 0.281i)T^{2} \)
7 \( 1 + (0.327 + 0.945i)T^{2} \)
13 \( 1 + (0.888 + 0.458i)T^{2} \)
17 \( 1 + (-0.928 + 0.371i)T^{2} \)
19 \( 1 + (0.327 - 0.945i)T^{2} \)
23 \( 1 + (1.49 - 0.770i)T + (0.580 - 0.814i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.273 - 1.12i)T + (-0.888 + 0.458i)T^{2} \)
37 \( 1 + (0.580 - 1.00i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.786 + 0.618i)T^{2} \)
43 \( 1 + (0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.0947 - 1.98i)T + (-0.995 - 0.0950i)T^{2} \)
53 \( 1 + (-1.02 + 1.18i)T + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (-1.50 - 0.442i)T + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.235 - 0.971i)T^{2} \)
71 \( 1 + (1.88 + 0.363i)T + (0.928 + 0.371i)T^{2} \)
73 \( 1 + (-0.235 - 0.971i)T^{2} \)
79 \( 1 + (-0.0475 - 0.998i)T^{2} \)
83 \( 1 + (-0.723 - 0.690i)T^{2} \)
89 \( 1 + (-0.396 + 0.254i)T + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.30393412419237005125394901311, −9.755277207654036323328407309148, −8.668682773251605571729176959340, −8.180064094400271202916431706287, −7.30189438108597929034861793393, −6.08004373719870834673966330451, −5.06124607466894114769593601700, −4.04673355936005820509714065576, −2.97972692322829054801260047072, −1.69652622325820957446652078205, 2.21291325523249749897468364674, 2.69022749268918652100679391326, 3.73623739423868995912344491494, 5.57604730032935121616953056154, 6.36300907112038696452044002062, 7.30467224016576386457079041865, 7.76477465511355849062073114378, 8.638445395538547711720980333698, 10.17610435853996224154714856022, 10.53987564089029339188971295828

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