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Properties

Label 2-738-123.101-c1-0-4
Degree $2$
Conductor $738$
Sign $-0.572 - 0.819i$
Analytic cond. $5.89295$
Root an. cond. $2.42754$
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  + (−0.453 − 0.891i)2-s + (−0.587 + 0.809i)4-s + (0.657 + 4.15i)5-s + (−1.16 + 0.993i)7-s + (0.987 + 0.156i)8-s + (3.39 − 2.47i)10-s + (−0.433 − 0.265i)11-s + (−4.02 + 0.316i)13-s + (1.41 + 0.585i)14-s + (−0.309 − 0.951i)16-s + (4.33 − 1.04i)17-s + (−2.00 − 0.157i)19-s + (−3.74 − 1.90i)20-s + (−0.0399 + 0.507i)22-s + (−0.985 + 3.03i)23-s + ⋯
L(s)  = 1  + (−0.321 − 0.630i)2-s + (−0.293 + 0.404i)4-s + (0.294 + 1.85i)5-s + (−0.439 + 0.375i)7-s + (0.349 + 0.0553i)8-s + (1.07 − 0.781i)10-s + (−0.130 − 0.0801i)11-s + (−1.11 + 0.0877i)13-s + (0.377 + 0.156i)14-s + (−0.0772 − 0.237i)16-s + (1.05 − 0.252i)17-s + (−0.460 − 0.0362i)19-s + (−0.837 − 0.426i)20-s + (−0.00851 + 0.108i)22-s + (−0.205 + 0.632i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $-0.572 - 0.819i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ -0.572 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327418 + 0.627851i\)
\(L(\frac12)\) \(\approx\) \(0.327418 + 0.627851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 + 0.891i)T \)
3 \( 1 \)
41 \( 1 + (0.957 - 6.33i)T \)
good5 \( 1 + (-0.657 - 4.15i)T + (-4.75 + 1.54i)T^{2} \)
7 \( 1 + (1.16 - 0.993i)T + (1.09 - 6.91i)T^{2} \)
11 \( 1 + (0.433 + 0.265i)T + (4.99 + 9.80i)T^{2} \)
13 \( 1 + (4.02 - 0.316i)T + (12.8 - 2.03i)T^{2} \)
17 \( 1 + (-4.33 + 1.04i)T + (15.1 - 7.71i)T^{2} \)
19 \( 1 + (2.00 + 0.157i)T + (18.7 + 2.97i)T^{2} \)
23 \( 1 + (0.985 - 3.03i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.85 + 0.445i)T + (25.8 + 13.1i)T^{2} \)
31 \( 1 + (5.70 + 7.85i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.03 + 0.753i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (2.66 - 1.35i)T + (25.2 - 34.7i)T^{2} \)
47 \( 1 + (1.61 - 1.89i)T + (-7.35 - 46.4i)T^{2} \)
53 \( 1 + (-0.390 + 1.62i)T + (-47.2 - 24.0i)T^{2} \)
59 \( 1 + (-3.44 - 1.11i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.47 - 6.81i)T + (-35.8 - 49.3i)T^{2} \)
67 \( 1 + (-11.4 + 7.04i)T + (30.4 - 59.6i)T^{2} \)
71 \( 1 + (6.67 - 10.8i)T + (-32.2 - 63.2i)T^{2} \)
73 \( 1 + (-9.21 - 9.21i)T + 73iT^{2} \)
79 \( 1 + (-11.9 + 4.94i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + 7.87iT - 83T^{2} \)
89 \( 1 + (-9.94 - 11.6i)T + (-13.9 + 87.9i)T^{2} \)
97 \( 1 + (-2.00 - 3.27i)T + (-44.0 + 86.4i)T^{2} \)
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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

−10.64731045572131615745673570701, −9.725383009715429028428099637973, −9.550507979190036577292986418779, −7.924504970918626506731664066660, −7.31342915674590834947997807652, −6.37322567693778604072724195222, −5.41972207370318837558433013484, −3.79872902892483553216995656635, −2.91296645926711110741727181566, −2.15088248426244147594347097971, 0.38678910108781194691346468385, 1.80857812861748055583117042804, 3.79013890107201296947662557099, 4.98453998790191088689983140251, 5.38203388919524119278353129762, 6.59440348582398743359209755435, 7.63482616812017355761080546661, 8.395197063589476101717324909746, 9.152546922346330116657921629451, 9.842097088117817592489667327360

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